Graph unique-maximum and conflict-free colorings

Graph unique-maxim um and conflict-free colorings P anagiotis Cheilaris Cen ter for Adv anced Studies in Mathematics Ben-Gurion Universit y panagiot@math.bgu.ac.il G ´ eza T´ oth ∗ R ´ en yi Institute Hungarian Academy of Sciences geza@ren yi.hu Abstract W e in vestigate the relationship b et ween t wo kinds of vertex colorings of graphs: unique- maxim um colorings and conflict-free colorings. In a unique-maximum coloring, the colors are ordered, and in ev ery path of the graph the maxim um color appe ars only once. In a conflict-free coloring, in every path of the graph there is a color that appears only once. W e also study computational complexity asp ects of conflict-free colorings and pro ve a completeness result. Finally , we improv e lo wer b ounds for those c hromatic num b ers of the grid graph. Keyw ords : unique-maxim um coloring, ordered coloring, vertex ranking, conflict-free coloring 1 In tro duction In this pap er we study tw o types of vertex colorings of graphs, b oth related to paths. The first one is the follo wing: Definition 1.1. A unique-maximum c oloring with r esp e ct to p aths of G = ( V , E ) with k colors is a function C : V → { 1 , . . . , k } such that for each path p in G the maximum color o ccurs exactly once on the v ertices of p . The minimum k for whic h a graph G has a unique-maximum coloring with k colors is called the unique-maximum chr omatic numb er of G and is denoted by χ um ( G ). Unique maximum colorings are known alternatively in the literature as or der e d c olorings or vertex r ankings . The problem of computing unique-maximum colorings is a well-kno wn and widely studied problem (see e.g. [11]) with many applications including VLSI design [12] and p ar al lel Cholesky de c omp osition of matrices [13]. The Cholesky decomp osition metho d is used in solving sparse linear systems Ax = b , whenever A is a symmetric n × n p ositive-definite matrix, and is faster than the more general LU decomp osition. In [13], giv en a symmetric n × n p ositive-definite matrix A , a graph G ( A ) on n v ertices is defined whic h enco des the data dep endencies b etw een differen t columns in the linear system. The unique-maximum c hromatic n umber of G ( A ) is a rough estimate of the work required in parallel Cholesky decomp osition of matrix A . The problem is also interesting for the Op erations Researc h communit y , b ecause it has applications in planning efficient assembly of pr o ducts in man ufacturing systems [10]. In general, it seems that the vertex ranking problem can mo del situations where interrelated tasks ha ve to b e accomplished fast in parallel (assembly from parts, parallel query optimization in databases, etc.) Another application of unique-maxim um colorings is in estimating the worst-case complexit y of finding lo c al optima in ∗ Supp orted b y OTKA T 038397 and 046246. 1 neighb orho o d structur es . A neigh b orho o d structure is a connected graph in whic h ev ery v ertex has a real v alue. Supp ose that we w ant to find a vertex v which is a lo cal optimum. F or example, if v is a local minimum, then its v alue is not greater than the v alues of its adjacen t vertices. The goal is to query as few vertices of the neigh b orho o d structure as p ossible. In some classes of b ounded-degree neigh b orho o d structures (lik e grids), the w orst-case complexit y of finding a lo cal optim um is closely related to the unique-maximum chromatic num b er of the corresp onding graph (see [14]). The other type of vertex coloring can b e seen as a relaxation of the unique-maxim um coloring. Definition 1.2. A c onflict-fr e e c oloring with r esp e ct to p aths of G = ( V , E ) with k colors is a function C : V → { 1 , . . . , k } such that for eac h path p in G there is a color that o ccurs exactly once on the v ertices of p . The minim um k for whic h a graph G has a conflict-free coloring with k colors is called the c onflict-fr e e chr omatic numb er of G and is denoted by χ cf ( G ). Conflict-free coloring of graphs with resp ect to paths is a sp ecial case of conflict-free colorings of hypergraphs, studied in Even et al. [8] and Smoro dinsky [18]. One of the applications of conflict- free colorings is that it represen ts a frequency assignmen t for cellular netw orks. A cellular netw ork consists of t wo kinds of no des: b ase stations and mobile agents . Base stations hav e fixed p ositions and pro vide the backbone of the net work; they are represen ted by vertices in V . Mobile agen ts are the clients of the netw ork and they are serv ed by base stations. This is done as follows: Ev ery base station has a fixed frequency; this is represen ted b y the coloring C , i.e., colors represent frequencies. If an agen t w ants to establish a link with a base station it has to tune itself to this base station’s frequency . Since agents are mobile, they can b e in the range of many different base stations. T o a void in terference, the system must assign frequencies to base stations in the following wa y: F or an y range, there m ust b e a base station in the range with a frequency that is not used b y some other base station in the range. One can solve the problem b y assigning n different frequencies to the n base stations. Ho wev er, using man y frequencies is exp ensive, and therefore, a scheme that reuses frequencies, where p ossible, is preferable. Conflict-free coloring problems hav e b een the sub ject of many recent pap ers due to their practical and theoretical in terest (see e.g. [15, 9, 6, 7, 3]). Most approaches in the conflict-free coloring literature use unique-maximum colorings (a notable exception is the ‘triples’ algorithm in [3]), b ecause unique-maxim um colorings are easier to argue ab out in proofs, due to their additional structure. Another adv an tage of unique-maximum colorings is the simplicit y of computing the unique color in an y range (it is alw ays the maxim um color), given a unique-maximum coloring, whic h can b e helpful if very simple mobile devices are used by the agen ts. F or general graphs, finding the exact unique-maximum chromatic n umber of a graph is NP- complete [17, 14] and there is a p olynomial time O (log 2 n ) approximation algorithm [5], where n is the n umber of vertices. Since the problem is hard in general, it makes sense to study sp ecific graphs. The m × m grid , G m , is the c artesian pr o duct of tw o paths, each of length m − 1, that is, the vertex set of G m is { 0 , . . . , m − 1 } × { 0 , . . . , m − 1 } and the edges are {{ ( x 1 , y 1 ) , ( x 2 , y 2 ) } | | x 1 − x 2 | + | y 1 − y 2 | ≤ 1 } . It is kno wn [11] that for general planar graphs the unique-maximum c hromatic num b er is O ( √ n ). Grid graphs are planar and therefore the O ( √ n ) b ound applies. One migh t exp ect that, since the grid has a relatively simple and regular structure, it should not be hard to calculate its unique-maximum chromatic num b er. This is why it is rather striking that, even though it is not hard to sho w upp er and lo wer b ounds that are only a small constant multiplicativ e factor apart, the exact v alue of these c hromatic num b ers is not kno wn, and has b een the sub ject of [1, 2]. 2 P ap er organization. In the rest of this section w e provide the necessary definitions and some earlier results. In section 2, we pro ve that it is coNP-complete to decide whether a given vertex coloring of a graph is conflict-free with resp ect to paths. In section 3, w e sho w that for every graph χ um ( G ) ≤ 2 χ cf ( G ) − 1 and provide a sequence of graphs for which the ratio χ um ( G ) /χ cf ( G ) tends to 2. In section 4, we in tro duce tw o games on graphs that help us relate the t wo c hromatic num b ers for the square grid graph. In section 5, we show a low er b ound on the unique-maximum c hromatic n umber of the square grid graph, impro ving previous results. Conclusions and op en problems are presen ted in section 6. 1.1 Preliminaries Definition 1.3. A graph X is a minor of Y , denoted as X 4 Y , if X can b e obtained from Y by a sequence of the following three op erations: vertex deletion, edge deletion, and edge contraction. Edge con traction is the pro cess of merging b oth endp oin ts of an edge into a new vertex, whic h is connected to all v ertices adjacent to the tw o endp oints. Given a unique-maxim um coloring C of Y , we get the induc e d c oloring of X as follo ws. T ak e a sequence of vertex deletions, edge deletions, and edge con tractions so that w e obtain X from Y . F or the v ertex and edge deletion op erations, just keep the colors of the remaining vertices. F or the edge con traction op eration, say along edge xy , whic h giv es rise to the new vertex v xy , set C 0 ( v xy ) = max( C ( x ) , C ( y )), and keep the colors of all other v ertices. Prop osition 1.4. [4] If X 4 Y , and C is a unique-maximum c oloring of Y , then the induc e d c oloring C 0 is a unique-maximum c oloring of X . Conse quently, χ um ( X ) ≤ χ um ( Y ) . The (traditional) chromatic num b er of a graph is denoted b y χ ( G ) and is the smallest num b er of colors in a vertex coloring for whic h adjacent vertices are assigned different colors. A simple relation b et ween the c hromatic num b ers w e hav e defined so far is the following. Prop osition 1.5. F or every gr aph G , χ ( G ) ≤ χ cf ( G ) ≤ χ um ( G ) . Pr o of. Since every unique-maxim um coloring is also a conflict-free coloring, w e hav e χ cf ( G ) ≤ χ um ( G ). A traditional coloring can be defined as a coloring in whic h paths of length one are conflict- free. Therefore ev ery conflict-free coloring is also a traditional coloring and th us χ ( G ) ≤ χ cf ( G ). Moreo ver, we prov e that b oth conflict-free and unique-maximum c hromatic num b ers are mono- tone under taking subgraphs. Prop osition 1.6. If X ⊆ Y , then χ cf ( X ) ≤ χ cf ( Y ) and χ um ( X ) ≤ χ um ( Y ) . Pr o of. T ak e the restriction of an y conflict-free or unique-maximum coloring of graph Y to the v ertex set V ( X ). This is a conflict-free or unique maxim um coloring of graph X , resp ectiv ely , b ecause the set of paths of graph X is a subset of all paths of Y . If v is a vertex (resp. S is a set of vertices) of graph G = ( V , E ), denote b y G − v (resp. G − S ) the graph obtained from G by deleting vertex v (resp. vertices of S ) and adjacen t edges. Definition 1.7. A subset S ⊆ V is a sep ar ator of a connected graph G = ( V , E ) if G − S is disconnected or empt y . A separator S is minimal if no prop er subset S 0 ⊂ S is a separator. 3 2 Deciding whether a coloring is conflict-free In this section, w e sho w a difference b et ween the t wo c hromatic num bers χ um and χ cf , from the com- putational complexit y asp ect. F or the notions of complexity classes, hardness, and completeness, w e refer, for example, to [16]. As w e mentioned b efore, in [17, 14], it is shown that computing χ um for general graphs is NP- complete. T o b e exact the follo wing problem is NP-complete: “Given a graph G and an in teger k , is it true that χ um ( G ) ≤ k ?”. The ab ov e fact implies that it is p ossible to c heck in p olynomial time whether a given coloring of a graph is unique-maximum with resp ect to paths. W e remark that b oth the conflict-free and the unique-maximum prop erties hav e to b e true in every path of the graph. Ho wev er, a graph with n v ertices can hav e exp onential in n num b er of distinct sets of v ertices, each one of whic h is a vertex set of a path in the graph. F or unique-maxim um colorings w e can find a shortcut as follows: Given a (connected) graph G and a vertex coloring of it, consider the set of vertices S of unique colors. Let u, v ∈ V \ S such that they b oth hav e the maximum color that app ears in V \ S . If there is a path in G − S from u to v , then this path violates the unique maxim um prop ert y . Therefore, S has to b e a separator in G , which can b e chec ked in p olynomial time, otherwise the coloring is not unique-maximum. If G − S is not empty , we can pro ceed analogously for eac h of its comp onen ts. F or conflict-free colorings there is no suc h shortcut, unless coNP = P, as the following theorem implies. Theorem 2.1. It is c oNP-c omplete to de cide whether a given gr aph and a vertex c oloring of it is c onflict-fr e e with r esp e ct to p aths. Pr o of. In order to pro ve that the problem is coNP-complete, we pro ve that it is coNP-hard and also that it b elongs to coNP . W e sho w coNP-hardness b y a reduction from the complemen t of the Hamiltonian path problem. F or every graph G , w e construct in polynomial time a graph G ∗ of p olynomial size together with a coloring C of its v ertices such that G has no Hamiltonian path if and only if C is conflict-free with resp ect to paths of G ∗ . Assume the v ertices of graph G are v 1 , v 2 , . . . , v n . Then, graph G ∗ consists of t wo isomorphic copies of G , denoted by ˆ G and ˇ G , with v ertex sets v 1 , v 2 , . . . , v n and v 1 , v 2 , . . . , v n , resp ectiv ely . Additionally , for every 1 ≤ i ≤ n , G ∗ con tains the path P i = v i , v i, 1 , v i, 2 , . . . , v i,i − 1 , v i,i +1 , . . . , v i,n , v i , where, for ev ery i , v i, 1 , v i, 2 , . . . , v i,i − 1 , v i,i +1 , . . . , v i,n are new v ertices. W e use the follo wing notation for the t wo p ossible directions to tra verse this path: P ↓ i = ( v i, 1 , . . . , v i,i − 1 , v i,i +1 , . . . , v i,n ) , P ↑ i = ( v i,n , . . . , v i,i +1 , v i,i − 1 , . . . , v i, 1 ) . W e call paths P i c onne cting p aths . W e now describ e the coloring of V ( G ∗ ). F or every i , we set C ( v i ) = C ( v i ) = i . F or every i > j , w e set C ( v i,j ) = C ( v j,i ) = n +  i − 1 2  + j . Observe that ev ery color o ccurs exactly in t wo v ertices of G ∗ . If G has a Hamiltonian path, say v 1 v 2 . . . v n , then there is a path through all vertices of G ∗ , either v 1 P ↓ 1 v 1 v 2 P ↑ 2 v 2 . . . v n − 1 P ↓ n − 1 v n − 1 v n P ↑ n v n , if n is even, 4 or v 1 P ↓ 1 v 1 v 2 P ↑ 2 v 2 . . . v n − 1 P ↑ n − 1 v n − 1 v n P ↓ n v n , if n is o dd. But then, this path has no uniquely o ccurring color and th us C is not conflict-free. Supp ose now that C is not a conflict-free coloring. W e prov e that G has a Hamiltonian path. By the assumption, there is a path P in G ∗ whic h is not conflict-free. This path must contain none or b oth v ertices of eac h color. Therefore, P can not b e completely contained in ˆ G , or in ˇ G , or in some P i . Also, P can not con tain only one of v i and v i , for some i . Therefore, P must contain b oth v i and v i for a non-empt y subset of indices i . Then, it must contain completely some P i , b ecause vertices in ˆ G and ˇ G can only b e connected with some complete P i . But since eac h one of the n − 1 colors of this P i o ccurs in a differen t connecting paths, P m ust con tain a v ertex in every connecting path. But then P must con tain ev ery v i and v i , b ecause vertices in P i can only b e connected to the rest of the graph through one of v i or v i . Supp ose that P is not a Hamiltonian path of G ∗ . Observe that if P do es not contain all v ertices of some connecting path P i , then one of its end v ertices should b e there. If P do es not contain v ertex v i,j , then it can not con tain v j,i either. But then one end v ertex of P should b e on P i , the other one on P j , and all other v ertices of G ∗ are on P . Therefore, we can extend P such that it con tains v i,j and v j,i as w ell. So assume in the sequel that P is a Hamiltonian path of G ∗ . No w we mo dify P , if necessary , so that b oth of its end-vertices e and f lie in V ( ˆ G ) ∪ V ( ˇ G ). If e and f are adjacent in G ∗ , then add the edge ef to P and we get a Hamiltonian cycle of G ∗ . No w remov e one of its edges which is either in ˆ G , or in ˇ G and get the desired Hamiltonian path. Supp ose now that e and f are not adjacent, and e is on one of the connecting paths. Then e should b e adjacen t to the end v ertex e 0 of that connecting path, whic h is in ˆ G or in ˇ G . Add edge ee 0 to P . W e get a cycle and a path joined in e 0 . Remov e the other edge of the cycle adjacent to e 0 . W e ha ve a Hamiltonian path no w, whose end v ertex is e 0 instead of e . Pro ceed analogously for f , if necessary . No w we hav e a Hamiltonian path P of G ∗ with end-v ertices in V ( ˆ G ) ∪ V ( ˇ G ). Then, P is in the form, sa y , v 1 P ↓ 1 v 1 v 2 P ↑ 2 v 2 . . . v n − 1 P ↓ n − 1 v n − 1 v n P ↑ n v n , if n is even, or v 1 P ↓ 1 v 1 v 2 P ↑ 2 v 2 . . . v n − 1 P ↑ n − 1 v n − 1 v n P ↓ n v n , if n is o dd. But then, v 1 v 2 . . . v n is a Hamiltonian path in G . Finally , the problem is in coNP b ecause one can verify that a coloring of a giv en graph is not conflict-free in p olynomial time, by giving the corresp onding path. W e show an example graph G , its transformation graph G ∗ , and its coloring C in figure 1. 3 The t w o c hromatic n um b ers of general graphs W e ha ve seen that χ um ( G ) ≥ χ cf ( G ) (proposition 1.5). In this section w e sho w that χ um ( G ) can not b e larger than an exp onen tial function of χ cf ( G ). W e also pro vide an infinite sequence of graphs H 1 , H 2 , . . . , for which lim k →∞ ( χ um ( H k ) /χ cf ( H k )) = 2. The path of n vertices is denoted b y P n . It is known that χ um ( P n ) = b log 2 n c + 1 (see for example [8]). Lemma 3.1. F or every p ath P n , χ cf ( P n ) = b log 2 n c + 1 . 5 v 1 v 2 v 3 v 4 v 1 v 2 v 3 v 4 v 1 v 2 v 3 v 4 v 1 , 2 v 1 , 3 v 1 , 4 v 2 , 1 v 2 , 3 v 2 , 4 v 3 , 1 v 3 , 2 v 3 , 4 v 4 , 1 v 4 , 2 v 4 , 3 1 2 3 4 1 2 3 4 5 6 8 5 7 9 6 7 10 8 9 10 Figure 1: Example graphs G , G ∗ , and coloring C of G ∗ Pr o of. By prop osition 1.5, χ cf ( P n ) ≤ χ um ( P n ). W e pro ve a matching lo wer b ound by induction. W e ha ve χ cf ( P 1 ) ≥ 1. F or n > 1, there is a uniquely o ccurring color in an y conflict-free coloring of the the whole path P n . Then, χ cf ( P n ) ≥ 1 + χ cf ( P b n/ 2 c ), which implies χ cf ( P n ) ≥ b log 2 n c + 1. Moreo ver, w e are going to use the following result (lemma 5.1 of [11]): If the longest path of G has k v ertices, then χ um ( G ) ≤ k . Prop osition 3.2. F or every gr aph G , χ um ( G ) ≤ 2 χ cf ( G ) − 1 . Pr o of. Set j = χ cf ( G ). F or any path P ⊆ G , χ cf ( P ) ≤ j , therefore, by lemma 3.1, the longest path has at most 2 j − 1 vertices, so by lemma 5.1 of [11], χ um ( G ) ≤ 2 j − 1. W e define recursively the following sequence of graphs: Graph H 0 is a single v ertex. Supp ose that w e ha ve already defined H k − 1 . Then H k consists of (a) a K 2 k +1 − 1 , (b) 2 k +1 − 1 copies of H k − 1 , and (c) for for eac h i , 1 ≤ i ≤ 2 k +1 − 1, the i -th vertex of the K 2 k +1 − 1 is connected by an edge to one of the vertices of the i -th copy of H k − 1 . H 0 H 1 H 2 · · · K 2 k +1 − 1 K 2 k +1 − 1 H k − 1 H k − 1 H k − 1 H k − 1 H k − 1 H k − 1 H k − 1 H k Figure 2: Sequence of graphs Lemma 3.3. F or k ≥ 0 , χ cf ( H k ) = 2 k +1 − 1 . Pr o of. By induction on k . F or k = 0, χ cf ( H 0 ) = 1. F or k > 0, we hav e H k ⊇ K 2 k +1 − 1 , therefore, χ cf ( H k ) ≥ 2 k +1 − 1. 6 In order to prov e that χ cf ( H k ) ≤ 2 k +1 − 1, it is enough to describ e a conflict-free coloring of H k with 2 k +1 − 1 colors, given a conflict-free coloring of H k − 1 with 2 k − 1 colors: W e color the v ertices of the clique K 2 k +1 − 1 with colors 1 , 2 , . . . , 2 k +1 − 1 suc h that the i -th v ertex is colored with color i . Consider these colors mo d 2 k +1 − 1, e. g. color 2 k +1 is iden tical to color 1. Recall that the i -th cop y of H k − 1 has a vertex connected to the i -th vertex of K 2 k +1 − 1 , and by induction we know that χ cf ( H k − 1 ) = 2 k − 1. Color the i -th cop y of H k − 1 , with colors i + 1 , i + 2 , . . . , i + 2 k − 1. W e claim that this vertex coloring of H k is conflict-free. If a path is completely contained in a copy of H k − 1 , then it is conflict-free by induction. If a path is completely con tained in the clique K 2 k +1 − 1 , then it is also conflict-free, b ecause all colors in the clique part are differen t. If a path con tains vertices from a single copy of H k − 1 , say , the i -th cop y , and the clique, then the i -th v ertex of the clique is on the path and uniquely colored. The last case is when a path contains v ertices from exactly t wo copies of H k − 1 . Supp ose that these are the i -th and j -th copies of H k − 1 , 1 ≤ i < j ≤ 2 k +1 − 1. If i + 2 k − 1 < j , then color j is unique in the path; indeed, the i -th cop y of H k − 1 is colored with colors i + 1 , . . . i + 2 k − 1, and the j -th copy of H k − 1 is colored with colors j + 1 , . . . j + 2 k − 1, while color j app ears only once in K 2 k +1 − 1 . Similarly , if i + 2 k − 1 ≥ j , then color i is unique in the path. Lemma 3.4. χ um ( H k ) ≤ 2 k +2 − k − 3 . Pr o of. By induction. F or k = 0, χ um ( H 0 ) = 1. F or k > 0, in order to color H k use the 2 k +1 − 1 differen t highest colors for the clique part. By the inductive hypothesis χ um ( H k − 1 ) ≤ 2 k +1 − k − 2. F or eac h copy of H k − 1 , use the same coloring with the 2 k +1 − k − 2 lo west colors. This coloring of H k is unique maxim um. Indeed, if a path is contained in a cop y of H k − 1 then it is unique maximum b y induction, and if it con tains a v ertex in the clique part, then it is also unique maximum. The total n umber of colors is 2 k +2 − k − 3. Lemma 3.5. If Y is a gr aph that c onsists of a K ` and ` isomorphic c opies of a c onne cte d gr aph X , such that for 1 ≤ i ≤ ` a vertex of it i -th c opy is c onne cte d to the i -th vertex of K ` by an e dge. Then we have χ um ( Y ) ≥ ` − 1 + χ um ( X ) Pr o of. By induction on ` . F or ` = 1, w e ha ve that χ um ( Y ) ≥ χ um ( X ), b ecause Y ⊇ X . F or the inductiv e step, for ` > 1, if Y consists of a K l and ` copies of X , then Y is connected, and thus con tains a v ertex v with unique color. But then, Y − v ⊇ Y 0 , where Y 0 is a graph that consists of a K ` − 1 and ` − 1 isomorphic copies of a X , each connected to a different vertex of K ` − 1 , and thus χ um ( Y ) = 1 + χ um ( Y 0 ) ≥ ` − 1 + χ um ( X ). Lemma 3.6. χ um ( H k ) ≥ 2 k +2 − 2 k − 3 . Pr o of. By induction. F or k = 0, χ um ( H 0 ) = 1. F or k > 0, by the inductive hypothesis and lemma 3.5, χ um ( H k ) ≥ 2 k +1 − 1 − 1 + 2 k +1 − 2( k − 1) − 3 = 2 k +2 − 2 k − 3 Theorem 3.7. We have lim k →∞ ( χ um ( H k ) /χ cf ( H k )) = 2 . Pr o of. F rom lemmas 3.3, 3.4, 3.6, we hav e 2 k +2 − 2 k − 3 2 k +1 − 1 ≤ χ um ( H k ) χ cf ( H k ) ≤ 2 k +2 − k − 3 2 k +1 − 1 whic h implies that the ratio tends to 2. 7 4 The t w o c hromatic n um b ers of a square grid In this section, w e define t wo games on graphs, eac h play ed by tw o pla yers. The first game char- acterizes completely the unique-maximum c hromatic num b er of the graph. The second game is related to the conflict-free chromatic n umber of the graph. W e use the tw o games to prov e that the conflict-free chromatic num b er of the square grid is a function of the unique-maxim um c hromatic n umber of the square grid. This is useful b ecause it allows to translate existing low er b ounds on the unique-maxim um c hromatic num b er of the square grid to lo wer b ounds on the corresp onding conflict-free c hromatic n umber. F or any graph G , and subset of its vertices V 0 ⊂ V ( G ), let G [ V 0 ] denote the subgraph of G induced by V 0 . The first game (which is pla yed on a graph G by t wo pla y ers) is the c onne cte d c omp onent game : i ← 0; G 0 ← G while V ( G i ) 6 = ∅ : incremen t i by 1 Pla yer 1 chooses a connected comp onen t S i of G i − 1 Pla yer 2 chooses a vertex v i ∈ S i G i ← G i − 1 [ S i \ { v i } ] The game is finite, b ecause if G i is not empt y , then G i +1 is a strict subgraph of G i . The result of the game is its length, that is, the final v alue of i . Play er 1 tries to mak e the final v alue of i as large as p ossible and thus is the maximizer play er. Pla yer 2 tries to mak e the final v alue of i as small as p ossible and th us is the minimizer play er. If b oth pla yers pla y optimally , then the result is the value of the connected comp onen t game on graph G , whic h is denoted by v cs ( G ). Prop osition 4.1. In the c onne cte d c omp onent game, ther e is a str ate gy for player 2 (the mini- mizer), so that the r esult of the game is at most χ um ( G ) , i.e., v cs ( G ) ≤ χ um ( G ) . Pr o of. By induction on χ um ( G ): If χ um ( G ) = 0, i.e., the graph is empty , the v alue of the game is 0. If χ um ( G ) = k > 0, then in the first turn some connected comp onent S 1 is chosen by play er 1. Then, the strategy of pla yer 2 is to take an optimal unique-maxim um coloring C of G and c ho ose a v ertex v 1 in S 1 that has a unique color in S 1 . Then, G 1 = G [ S 1 \ { v 1 } ] ⊂ G 0 and the restriction of C to S 1 \ { v 1 } is a unique-maximum coloring of G 1 that is using at most k − 1 colors. Th us, χ um ( G 1 ) ≤ k − 1, and by the inductive h yp othesis pla yer 2 has a strategy so that the result of the game on G 1 is at most k − 1. Therefore, play er 2 has a strategy so that the result of the game on G 0 = G is at most 1 + k − 1 = k . Lemma 4.2. F or every v ∈ V ( G ) , χ um ( G − v ) ≥ χ um ( G ) − 1 Pr o of. Assume for the sake of contradiction that there exists a v ∈ V ( G ) for which χ um ( G − v ) < χ um ( G ) − 1. Then an optimal coloring of G − v can b e extended to a coloring of G , where v has a new unique maximum color. Therefore there is a coloring of G that uses less than χ um ( G ) − 1 + 1 = χ um ( G ) colors; a contradiction. Prop osition 4.3. In the c onne cte d c omp onent game, ther e is a str ate gy for player 1 (the maxi- mizer), so that the r esult of the game is at le ast χ um ( G ) , i.e., v cs ( G ) ≥ χ um ( G ) . Pr o of. By induction on χ um ( G ): If χ um ( G ) = 0, i.e., the graph is empty , the result of the game is zero. If χ um ( G ) = k > 0, the strategy of pla yer 1 is to choose a connected comp onent S 1 suc h that χ um ( G [ S 1 ]) = k . F or ev ery choice of v 1 b y Play er 2, by lemma 4.2, χ um ( G 1 ) ≥ k − 1, and thus, by the inductive hypothesis play er 1 has a strategy so that the result of the game on G 1 is at least k − 1. Therefore, the result of the game on G 0 = G is at least 1 + k − 1 = k . 8 Corollary 4.4. F or every gr aph, v cs ( G ) = χ um ( G ) . The second game (also play ed on a graph G b y tw o play ers) is the p ath game : i ← 0; G 0 ← G while V ( G i ) 6 = ∅ : incremen t i by 1 Pla yer 1 chooses the set of v ertices S i of a path of G i − 1 Pla yer 2 chooses a vertex v i ∈ S i G i ← G i − 1 [ S i \ { v i } ] The only difference with the connected comp onen t game is that in the path game the vertex set S i that maximizer c ho oses is the vertex set of a path of the graph G i − 1 . If b oth pla yers play optimally , then the result is the value of the path game on graph G , whic h is denoted b y v p ( G ). Prop osition 4.5. In the p ath game, ther e is a str ate gy for player 2 (the minimizer), so that the r esult of the game is at most χ cf ( G ) , i.e., v p ( G ) ≤ χ cf ( G ) . Pr o of. By induction on χ cf ( G ): If χ cf ( G ) = 0, i.e., the graph is empt y , the v alue of the game is 0. If χ cf ( G ) = k > 0, then in the first turn some v ertex set S 1 of a path of G is c hosen by pla yer 1. Then, the strategy of play er 2 is to find an optimal conflict-free coloring C of G and choose a vertex v 1 in S 1 that has a unique color in S 1 . Then, G 1 = G [ S 1 \ { v 1 } ] ⊂ G 0 and the restriction of C to S 1 \ { v 1 } is a conflict-free coloring of G 1 that is using at most k − 1 colors. Thus, χ cf ( G 1 ) ≤ k − 1, and b y the inductiv e hypothesis pla yer 2 has a strategy so that the result of the game is at most k − 1. Therefore, play er 2 has a strategy so that the result of the game is at most 1 + k − 1 = k . A prop osition analogous to 4.3 for the path game is not true. F or example, for the complete binary tree of four lev els (with 15 vertices, 8 of which are leav es), B 4 , it is not difficult to c heck that v p ( B 4 ) = v p ( P 7 ) = 3, but χ cf ( B 4 ) = 4. No w, we are going to concen trate on the square grid graph. Assume that m is ev en. W e intend to translate a strategy of play er 1 (the maximizer) on the connected comp onen t game for graph G m/ 2 to a strategy for play er on the path game for graph G m . Observ e that for every connected graph G , there is an ordering of its v ertices, v 1 , v 2 , . . . , v n suc h that the subgraph induced by the first k v ertices (for every 1 ≤ k ≤ n ) is also connected. Just pick a vertex to b e v 1 , and add the other vertices one b y one such that the new vertex v i is connected to the graph induced by v 1 , . . . , v i − 1 . This is p ossible, since G itself is connected. W e call such an ordering of the vertices an always-c onne cte d or dering . No w we decomp ose the v ertex set of G m in to groups of four v ertices, Q x,y = { (2 x, 2 y ) , (2 x + 1 , 2 y ) , (2 x, 2 y + 1) , (2 x + 1 , 2 y + 1) } , for 0 ≤ x, y < m/ 2, called sp e cial quadruples , or briefly quadruples. W e denote the set of quadruples with W m = { Q x,y | 0 ≤ x, y < m/ 2 } and let τ ( x, y ) = Q x,y b e a bijection betw een vertices of V ( G m/ 2 ) and W m . Extend τ for subsets of vertices of G m/ 2 in a natural wa y , for any S ⊆ V ( G m/ 2 ), τ ( S ) = S ( x,y ) ∈ S τ ( x, y ). Define also a kind of in verse τ 0 of τ as τ 0 ( x, y ) = ( b x/ 2 c , b y / 2 c ) for any 0 ≤ x, y < m , and for any S ⊆ V ( G m ), τ 0 ( S ) = { τ 0 ( x, y ) | ( x, y ) ∈ S } . Let ( x, y ) ∈ V ( G m/ 2 ). W e call vertices ( x, y + 1), ( x, y − 1), ( x − 1 , y ), and ( x + 1 , y ), if they exist, the upp er , lower , left , and right neighb ors of ( x, y ), resp ectively . Similarly , quadruples Q x,y +1 , Q x,y − 1 , Q x − 1 ,y , and Q x +1 ,y the upp er , lower , left , and right neighb ors of Q x,y , resp ectiv ely . Quadruple Q x,y induces four edges in G m , { (2 x + 1 , 2 y ) , (2 x + 1 , 2 y + 1) } , { (2 x, 2 y ) , (2 x, 2 y + 1) } , { (2 x, 2 y ) , (2 x, 2 y + 1) } , { (2 x + 1 , 2 y ) , (2 x + 1 , 2 y + 1) } , we call them upp er , lower , left , and right edges of Q x,y . 9 By dir e ction d , w e mean one of the four basic directions, up , down , left , right . F or a giv en set S ⊆ V ( G m/ 2 ), w e say that v ∈ S is op en in S in direction d , if its neigh b or in direction d is not in S . In this case we also say that τ ( v ) is op en in τ ( S ) in direction d . Lemma 4.6. If S induc es a c onne cte d sub gr aph in G m/ 2 , then ther e is a p ath in G m whose vertex set is τ ( S ) . Pr o of. W e prov e a stronger statemen t: If S induces a connected subgraph in G m/ 2 , then there is a cycle C in G m whose vertex set is τ ( S ), and if v ∈ S is op en in direction d in S , then C con tains the d -edge of τ ( v ). The pro of is b y induction on | S | = k . F or k = 1, τ ( S ) is one quadruple and w e can tak e its four edges. Supp ose that the statement has b een pro ved for | S | < k , and assume that | S | = k . Consider an alw ays-connected ordering v 1 , v 2 , . . . , v k of S . Let S 0 = S \ v k . By the induction h yp othesis, there is a cycle C 0 satisfying the requirements. V ertex v k has at least one neigh b or in S 0 , sa y , v k is the neigh b or of v i in direction d . But then, v i is op en in direction d in S 0 , therefore, C 0 con tains the d -edge of τ ( v i ). Remov e this edge from C 0 and substitute b y a path of length 5, passing through all four vertices of τ ( v k ). The resulting cycle, C , con tains all vertices of τ ( S ), it con tains each edge of τ ( v k ), except the one in the opposite direction to d , and it contains all edges of C 0 , except the d -edge of τ ( v k ), but v k is not open in S in direction d . This concludes the induction step, and the pro of. Prop osition 4.7. F or every m > 1 , v p ( G m ) ≥ v cs ( G b m/ 2 c ) . Pr o of. Assume, without loss of generality that m is ev en (if not w ork with graph G m − 1 instead). In order, to pro ve that v p ( G m ) ≥ v cs ( G b m/ 2 c ) it is enough, given a strategy for play er 1 in the connected set game for G m/ 2 , to construct a strategy for play er 1 (the maximizer) in the path game for G m , so that the result of the path game is at least as m uch as the result of the connected set game. W e present the argument as if play er 1, apart from the path game, plays in parallel a connected set game on G m/ 2 (for whic h pla yer 1 has a giv en strategy to c ho ose connected sets in ev ery round), where play er 1 also chooses the mov es of play er 2 in the connected set game. A t round i of the path game on G m , pla yer 1 sim ulates round i of the connected set game on G m/ 2 . At the start of round i , play er 1 has a graph G i − 1 ⊆ G m in the path game and a graph ˆ G i − 1 ⊆ G m/ 2 in the connected set game. Play er 1 c ho oses a set ˆ S i in the sim ulated connected set game from his given strategy , and then constructs the path-spanned set S i = τ ( ˆ S i ) (b y lemma 4.6) and pla ys it in the path game. Then pla yer 2 chooses a vertex v i ∈ S i . Pla yer 1 computes ˆ v i = τ 0 ( v i ) and simulates the mov e ˆ v i of play er 2 in the connected set game. This is a legal mov e for pla yer 2 in the connected set game b ecause ˆ v i ∈ ˆ S i . W e just hav e to pro ve that S i = τ ( ˆ S i ) is a legal mo ve for play er 1 in the path game, i.e., S i ⊆ V ( G i − 1 ). W e also hav e to prov e S i = τ ( ˆ S i ) is spanned b y a path in G i − 1 but this is alw ays true by lemma 4.6, since ˆ S i is a connected vertex set in ˆ G i − 1 . Since S i ⊆ τ ( V ( ˆ G i − 1 )), it is enough to prov e that at round i , τ ( V ( ˆ G i − 1 )) ⊆ V ( G i − 1 ). The pro of is b y induction on i . F or i = 1, G 0 = G m , ˆ G 0 = G m/ 2 , and th us τ ( V ( ˆ G 0 )) = V ( G 0 ). At the start of round i with i > 1, τ ( V ( ˆ G i − 1 )) ⊆ V ( G i − 1 ), by the inductive h yp othesis. Then, τ ( ˆ S i ) = S i and τ ( ˆ S i \ { ˆ v i } ) = τ ( ˆ S i ) \ τ ( ˆ v i ) = S i \ τ ( ˆ v i ) ⊆ S i \ { v i } , b ecause v i ∈ τ ( ˆ v i ). Thus, τ ( V ( ˆ G i − 1 [ ˆ S i \ { ˆ v i } ])) ⊆ V ( G i − 1 [ S i \ { v i } ]), i.e., τ ( V ( ˆ G i )) ⊆ V ( G i ). Theorem 4.8. F or every m > 1 , χ cf ( G m ) ≥ χ um ( G b m/ 2 c ) . 10 Pr o of. By prop osition 4.5, χ cf ( G m ) ≥ v p ( G m ), b y prop osition 4.7, v p ( G m ) ≥ v cs ( G b m/ 2 c ), and by prop osition 4.3, v cs ( G b m/ 2 c ) ≥ χ um ( G b m/ 2 c ). 5 Lo w er b ounds on the c hromatic n um b ers of the square grid Recall that G m is the m × m grid graph, that is, the cartesian pro duct of t wo paths, each of length m − 1. It was shown in [2] that χ um ( G m ) ≥ 3 m/ 2. The b est known upp er b ound is χ um ( G m ) ≤ 2 . 519 m , from [1, 2]. The main result of this section is the following improv ement of the lo wer b ound. Theorem 5.1. F or m ≥ 2 , χ um ( G m ) ≥ 5 3 m − 18 log 2 m . Pr o of. F or an y subset A ⊂ V ( G ), let N G ( A ) denote the b oundary of A , that is, all vertices whic h are not in A , but neigh b ors of some v ertex in A . Observe that in a unique-maximum coloring of a connected graph G , the set of v ertices of unique colors form a separator (see, e.g., [11]). Indeed, remo ve all vertices of unique colors from G , let G 0 b e the remaining graph and let color c b e the highest remaining color. It is not unique, let u and v b e tw o vertices of color c . Then there can not b e a path in G 0 from u to v , therefore, G 0 is not connected. W e will use induction on m . Consider a unique-maximum coloring of G m and take a minimal separator, formed by v ertices of unique colors. Using the separator and the coloring, after applying a carefully selected sequence of minor op erations (v ertex deletion, edge deletion, edge con traction) on G m , w e obtain an induced unique-maximum coloring (see definition 1.3) of G m 0 for some m 0 < m , and w e apply the induction h yp othesis to prov e the low er b ound. Throughout the proof, we consider G = G m in its standar d dr awing , that is, the v ertices are p oin ts ( x, y ), 0 ≤ x, y ≤ m − 1, tw o vertices ( x, y ) and ( x 0 , y 0 ) are connected if and only if | x − x 0 | + | y − y 0 | = 1, and edges are drawn as straight line segmen ts. If it is clear from the con text, w e do not make any notational distinction b etw een vertices (edges) and p oints (resp. segments) represen ting them. Denote by V the v ertices of the grid, that is, V = V ( G ). T ake an additional v ertex v , “outside” G m , say , at ( − 2 , − 2), and connect it with all b oundary vertices of G m , so that w e do not create any edge crossing. Let G 0 = G 0 m denote the resulting graph, Let V 0 = V ( G 0 ). Define graph H 0 and its dra wing as follows. The vertex set of H 0 is V 0 . V ertex v is connected to the b oundary v ertices of the grid, just like in G 0 . Two v ertices, ( x, y ) and ( x 0 , y 0 ) in the grid are connected b y a straight line segmen t in H 0 if and only if | x − x 0 | ≤ 1 and | y − y 0 | ≤ 1. Supp ose that S ⊂ V 0 , and H 0 [ S ] con tains a non-self-intersecting cycle C . Let A (resp. B ) b e those vertices in V 0 whic h are inside (resp. outside ) C . If A, B 6 = ∅ , then C is called a sep ar ating cycle. If A = ∅ , then C is called an empty cycle. Supp ose that C is a separating cycle. Since edges of H 0 and edges of G 0 do not in tersect each other, S separates A and B in G 0 . Supp ose now that S is a separator in G 0 and let A b e the vertex set of one of the connected comp onen ts, separated b y S . Clearly , the boundary of A , N G 0 ( A ) b elongs to S , and an easy case analysis sho ws that the edges of H 0 [ N G 0 ( A )], in the present drawing, separate the vertices of A from the other v ertices. Supp ose from now that S is a minimal separator. Then, by the previous observ ations, H 0 [ S ] con tains one or more separating cycles. Let C b e a separating cycle in H 0 [ S ] with the smallest n umber of p oin ts inside, and let A b e the set of these p oin ts. Then N G 0 ( A ) ⊂ C , but since N G 0 ( A ) already separates A from the other vertices, N G 0 ( A ) = S . Observe that the only empt y cycle in H 0 is the righ t angled triangle with leg 1. If H 0 [ S ] contains suc h a cycle, then one of its v ertices can b e remov ed from S and we still hav e a separator. Therefore, there are no empty cycles in H 0 [ S ]. Moreov er, b y the minimality of S , every separating cycle in H 0 [ S ] con tains exactly the p oin ts of A in its interior. It follows, that H 0 [ S ] is a cycle that has A in its interior, and the remaining p oin ts, V 0 \ ( S ∪ A ) in the exterior. 11 v v Figure 3: A cycle-separator in G , G 0 , and H 0 , for m = 7 v v Figure 4: A path-separator in G , G 0 , and H 0 , for m = 7 It is easy to see that if S is a separator in G m , then S ∪ { v } is a separator in G 0 m . On the other hand, if S is a separator is G 0 m , then S \ { v } is a separator in G 0 m . Consequently , if S is a minimal separator in G m , then either S is a minimal separator in G 0 m , or S ∪ { v } is a minimal separator in G 0 m . In the first case we sa y that S is a cycle-sep ar ator (see figure 3), in the second case we sa y that it is a p ath-sep ar ator (see figure 4) of G m . The vertices of a cycle-separator form a cycle in H 0 , and the v ertices of a path-separator form a path, whose first and last vertices are the only neigh b ors of v , that is, they are on the b oundary of the grid, and the other vertices of S are not on the b oundary . Our b ound is negative for m ≤ 64, so assume that m > 64, and the statement has b een prov ed for smaller v alues of m . Consider an optimal coloring of G m , and let S b e a minimal separator, all of whose v ertices hav e unique colors. Case 1: S is a cycle-sep ar ator. Let z b e the smallest v alue of x + y o ver all vertices of S , and let ( x, y ) b e the v ertex of S for which x + y = z , and y is the largest. Then vertex ( x + 1 , y − 1) is also a v ertex of S , and one of ( x, y + 1), ( x + 1 , y + 1) is also in S . Let ( x 0 , y 0 ) b e the v ertex of S for whic h x + y = z , and y is the smallest. Then y 0 < y , since ( x + 1 , y − 1) is in S . Moreo ver, v ertex ( x 0 − 1 , y 0 + 1) is also a vertex of S , and one of ( x 0 + 1 , y 0 ), ( x 0 + 1 , y 0 + 1) is also in S . Consider the follo wing con tractions of horizontal edges: ( x, m − 1)( x + 1 , m − 1), ( x, m − 2)( x + 1 , m − 2), . . . , ( x, y )( x + 1 , y ), ( x + 1 , y − 1)( x + 2 , y − 1), ( x + 2 , y − 2)( x + 3 , y − 2), . . . , ( x 0 , y 0 )( x 0 + 1 , y 0 ), ( x 0 , y 0 − 1)( x 0 + 1 , y 0 − 1), . . . , ( x 0 , 0)( x 0 + 1 , 0), and v ertical edges: (0 , y )(0 , y + 1), (1 , y )(1 , y + 1), . . . , ( x, y )( x, y + 1), ( x + 1 , y )( x + 1 , y + 1), ( x + 2 , y − 1)( x + 2 , y ), . . . , ( x 0 + 1 , y 0 )( x 0 + 1 , y 0 + 1), ( x 0 + 2 , y 0 )( x 0 + 2 , y 0 + 1), . . . , ( m − 1 , y 0 )( m − 1 , y 0 + 1). W e obtain a graph, whic h contains G m − 1 as a subgraph and the induced coloring uses at least tw o less colors that the coloring of G m See figure 5, where for each gray area, v ertices are contracted to a single v ertex. The induced coloring uses at least χ um ( G m − 1 ) colors, therefore, we ha ve χ um ( G m ) ≥ χ um ( G m − 1 ) + 2 ≥ 5 3 ( m − 1) − 18 log 2 ( m − 1) + 2 > 5 3 m − 18 log 2 m . 12 ( x, y ) ( x 0 , y 0 ) Figure 5: Graph G m with edge con tractions and its minor con taining G m − 1 Case 2: S is a p ath-sep ar ator. By symmetry we can assume that the path starts in column x = 0. If it ends in x = 0, y = 0, or in y = m − 1, then, we can remov e column x = 0, and either row y = 0 or y = m − 1, and get a unique maximum coloring of G m − 1 with at least tw o less colors. Then w e apply induction as in case 1. So we can assume that S ends in x = m − 1. It follows that | S | ≥ m . W e distinguish tw o sub cases. Sub case 2.1. S starts in x = 0 , ends in x = m − 1 , and | S | > m . Orien t the path formed b y the vertices of S . F or simplicity , call the orien ted path v 1 , . . . v | S | also S . The edges of S can b e of eigh t types, left, righ t, upp er, lo wer, left-upp er, left-low er, righ t-upp er, righ t-low er. Supp ose first that S con tains t wo edges, one of them is vertical (left or right edge), one of them is horizontal (upp er or low er edge), say , ( x, y )( x + 1 , y ) and ( x 0 , y 0 )( x 0 , y 0 + 1). Then con tract all edges ( x, i )( x + 1 , i ), and all edges ( i, y 0 )( i, y 0 + 1), 0 ≤ i ≤ m − 1, to obtain G m − 1 , whose induced coloring uses at most χ um ( G m ) − 2 colors. Therefore, w e hav e χ um ( G m ) ≥ χ um ( G m − 1 ) + 2 ≥ 5 3 ( m − 1) − 18 log 2 ( m − 1) + 2 > 5 m/ 3 − 18 log 2 m . So, we can assume in the sequel that either there are no v ertical edges, or no horizon tal edges in S . Supp ose that there are no horizontal edges, and let v i = ( x, y ) b e a vertex of S where y is the largest. Then v i − 1 v i is an upp er-righ t edge, and v i v i +1 is a lo wer-righ t edge, or v i − 1 v i is an upp er-left edge, and v i v i +1 is a lo wer-left edge. W e can assume the first one, otherwise we can take the opp osite orientation of S . Let v i , . . . , v j b e a maximal interv al of S where all edges are low er-righ t. By assumption, edge v j v j +1 can not b e horizon tal, b y the minimality of S it can not b e upp er, if it is low er, or low er-left, then we can pro ceed just like in the case of cycle-separators, by a sequence of edge con tractions w e can obtain an induced coloring of G m with t wo less colors and we are done b y induction. So, v j v j +1 can only b e an upp er-right edge. W e can apply the same argumen t for the next maximal in terv al v j , . . . , v k and obtain that v k v k +1 is a low er-righ t edge. W e can argue similarly “backw ards” on S , if v l , . . . , v i is a maximal interv al of upp er-right edges, then v l − 1 v l is a low er-right edge. It follows, that all edges of S are either upp er-right, or low er-right. But then S can not ha ve more then m vertices, a con tradiction. In the case when there are no vertic al edges, the argumen t is almost exactly the same. Sub case 2.2. S starts in x = 0 , ends in x = m − 1 , and | S | = m . Since If | S | = m , S = v 1 , v 2 , . . . , v m suc h that v i = ( i − 1 , y i ), for ev ery i . W e show that G m \ S con tains a subgraph isomorphic to G 2 k . 13 A B v i v i Figure 6: The sub case | S | = m Supp ose that 5 k ≤ m ≤ 5 k + 4. Consider the set of v ertices A = { ( x, y ) | k ≤ x ≤ 4 k − 1, 0 ≤ y ≤ 2 k − 1 } . Set A induces a 3 k × 2 k grid graph, G 3 k, 2 k , in G m . If A ∩ S = ∅ , then G m − S ⊇ G 3 k, 2 k ⊇ G 2 k ; otherwise some v i ∈ S b elongs to A , i.e., v i = ( i, y i ) with k ≤ i ≤ 4 k − 1 and 0 ≤ y i ≤ 2 k − 1. Then, consider the set of vertices B v i = { ( x, y ) | i − k + 1 ≤ x ≤ i + k , 3 k ≤ y ≤ m − 1 } , whic h contains a G 2 k subgraph in G m and it is disjoin t from S . Therefore, G m − S contains a subgraph isomorphic to G 2 k , and th us χ um ( G m ) ≥ m + χ um ( G 2 k ) ≥ m + 10 3 k − 18 log 2 2 k ≥ 5 3 m − 18 log 2 m . R emark 5.2 . By a slightly more careful calculation w e could get χ um ( G m ) ≥ 5 3 m − log 5 / 2 m . An immediate corollary from theorem 4.8 is the following. Corollary 5.3. F or m ≥ 2 , χ cf ( G m ) ≥ 5 6 m − 10 log 2 m . 6 Discussion and op en problems As we mentioned in the in tro duction, conflict-free and unique-maxim um colorings can be defined for h yp ergraphs. In the literature of conflict-free colorings, h yp ergraphs that are induced by geometric shap es hav e b een in the fo cus. It w ould b e interesting to show p ossible relations of the resp ective c hromatic num b ers in this setting. An interesting op en problem is to determine the exact v alue of the unique-maxim um chromatic n umber for the square grid G m . In this pap er, we improv ed the low er b ound asymptotically to 5 m/ 3, and w e believe that this b ound is still far from optimal. Observ e that in eac h case, our recursion step w ould allo w us to pro ve a low er bound of the form 2 m − o ( m ), with the exception of the last case, when | S | = m , that is the “b ottlenec k” of the pro of. W e b elieve that using a more complicated recursion, with grids of rectangular shap es, could lead to an impro vemen t. 14 Another area for improv emen t is the relation b etw een the tw o c hromatic num b ers for general graphs. W e hav e only found graphs which hav e unique-maximum chromatic n umber ab out twice the conflict-free c hromatic n umber, but the only b ound w e hav e pro v ed on χ um ( G ) is exp onen tial in χ cf ( G ). Finally , the coNP-completeness of chec king whether a coloring is conflict-free, implies that the decision problem for the conflict-free chromatic n umber is in complexity class Π p 2 (at the second lev el of the p olynomial hierarc hy). An in teresting direction for research w ould b e to attempt a pro of of Π p 2 completeness for this last decision problem. References [1] Amotz Bar-Noy , Panagiotis Cheilaris, Michael Lampis, V alia Mitsou, and Stathis Zac hos. Ordered coloring grids and related graphs. In Pr o c e e dings of the 16th International Col lo quium on Structur al Information and Communic ation Complexity (SIROCCO) , 2009. [2] Amotz Bar-Noy , Panagiotis Cheilaris, Michael Lampis, V alia Mitsou, and Stathis Zac hos. Ordered coloring of grids and related graphs. Submitted to a journal, 2009. [3] Amotz Bar-No y , Panagiotis Cheilaris, and Shakhar Smoro dinsky . Deterministic conflict-free coloring for in terv als: from offline to online. ACM T r ansactions on A lgorithms , 4(4):44:1–44:18, 2008. [4] Hans L. Bo dlaender, Jitender S. Deogun, Klaus Jansen, T on Kloks, Dieter Kratsch, Haik o M ¨ uller, and Zsolt T uza. Rankings of graphs. SIAM Journal on Discr ete Mathematics , 11(1):168–181, 1998. [5] Hans L. Bo dlaender, John R. Gilb ert, Hj´ alm tyr Hafsteinsson, and T on Kloks. Approximat- ing treewidth, path width, fron tsize, and shortest elimination tree. Journal of Algorithms , 18(2):238–255, 1995. [6] Ke Chen, Amos Fiat, Haim Kaplan, Meital Levy , Ji ˇ r ´ ı Matou ˇ sek, Elc hanan Mossel, J´ anos P ach, Mic ha Sharir, Shakhar Smoro dinsky , Uli W agner, and Emo W elzl. Online conflict-free coloring for in terv als. SIAM Journal on Computing , 36(5):1342–1359, 2007. [7] Khaled Elbassioni and Nabil H. Mustafa. Conflict-free colorings of rectangles ranges. In Pr o c e e dings of the 23r d International Symp osium on The or etic al Asp e cts of Computer Scienc e (ST ACS) , pages 254–263, 2006. [8] Guy Even, Zvi Lotker, Dana Ron, and Shakhar Smoro dinsky . Conflict-free colorings of sim- ple geometric regions with applications to frequency assignment in cellular netw orks. SIAM Journal on Computing , 33:94–136, 2003. [9] Sariel Har-P eled and Shakhar Smoro dinsky . Conflict-free coloring of p oints and simple regions in the plane. Discr ete and Computational Ge ometry , 34:47–70, 2005. [10] Anan th V. Iyer, H. Ronald Ratliff, and Gopalakrishanan Vijay an. Optimal no de ranking of trees. Information Pr o c essing L etters , 28:225–229, 1988. [11] Meir Katc halski, William McCuaig, and Suzanne Seager. Ordered colourings. Discr ete Math- ematics , 142:141–154, 1995. 15 [12] Charles E. Leiserson. Area-efficient graph lay outs (for VLSI). In Pr o c e e dings of the 21st Annual IEEE Symp osium on F oundations of Computer Scienc e (FOCS) , pages 270–281, 1980. [13] Joseph W.H. Liu. The role of elimination trees in sparse factorization. SIAM Journal on Matrix Analysis and Applic ations , 11(1):134–172, 1990. [14] Donna Crystal Llew ellyn, Craig A. T ov ey , and Mic hael A. T ric k. Lo cal optimization on graphs. Discr ete Applie d Mathematics , 23(2):157–178, 1989. [15] J´ anos Pac h and G´ eza T´ oth. Conflict free colorings. In Discr ete and Computational Ge ometry, The Go o dman-Pol lack F estschrift , pages 665–671. Springer V erlag, 2003. [16] Christos Papadimitriou. Computational Complexity . Addison W esley , 1993. [17] Alex P othen. The complexity of optimal elimination trees. T echnical Rep ort CS-88-16, De- partmen t of Computer Science, Pennsylv ania State Universit y , 1988. [18] Shakhar Smoro dinsky . Combinatorial Pr oblems in Computational Ge ometry . PhD thesis, Sc ho ol of Computer Science, T el-Aviv Universit y , 2003. 16