Optimal k-fold colorings of webs and antiwebs

Optimal k -fold colorings of w ebs and an tiw ebs ∗ Mano el Camp ˆ elo † a , Ricardo C. Corr ˆ ea ‡ b , Phablo F. S. Moura § c , and Marcio C. San tos ¶ b a Univ ersidade F ederal do Cear´ a, Departamento de Estat ´ ıstica e Matem´ atica Aplicada, Campus do Pici, Blo co 910, 60440-554 F ortaleza - CE, Brazil b Univ ersidade F ederal do Cear´ a, Departamento de Computa¸ c˜ ao, Campus do Pici, Blo co 910, 60440-554 F ortaleza - CE, Brazil c Univ ersidade de S˜ ao P aulo, Instituto de Matem´ atica e Estat ´ ıstica, Rua do Mat˜ ao 1010, 05508-090 S˜ ao P aulo - SP , Brazil No vem b er 27, 2024 Abstract A k -fold x -coloring of a graph is an assignment of (at least) k distinct colors from the set { 1 , 2 , . . . , x } to eac h vertex suc h that any t wo adjacent v ertices are assigned disjoin t sets of colors. The smallest n umber x such that G admits a k -fold x -coloring is the k -th chromatic n umber of G , denoted b y χ k ( G ). W e determine the exact v alue of this parameter when G is a w eb or an an tiweb. Our results generalize the known corresp onding results for o dd cycles and imply necessary and sufficient conditions under whic h χ k ( G ) attains its lo wer and upp er b ounds based on the clique, the fractional c hromatic and the c hromatic n umbers. Additionally , w e extend the concept of χ - critical graphs to χ k -critical graphs. W e iden tify the webs and antiw ebs having this prop ert y , for every integer k ≥ 1. Keyw ords: ( k -fold) graph coloring, (fractional) c hromatic n umber, clique and stable set num b ers, w eb and antiw eb ∗ A short v ersion of this pap er was presen ted at Simp´ osio Br asileir o de Pesquisa Op er acional, 2011. This w ork is partially supp orted by a CNPq/FUNCAP Pronem pro ject. † P artially supp orted b y CNPq-Brazil. mcampelo@lia.ufc.br ‡ correa@lia.ufc.br § P artially supp orted by CNPq-Brazil. Most of this work was done while the author was affiliated to Univ ersidade F ederal do Cear´ a. phablo@ime.usp.br ¶ P artially supp orted b y Cap es-Brazil. marciocs5@lia.ufc.br 1 In tro duction F or an y in tegers k ≥ 1 and x ≥ 1, a k -fold x -c oloring of a graph is an assignmen t of (at least) k distinct colors to each v ertex from the set { 1 , 2 , . . . , x } such that any t wo adjacen t vertices are assigned disjoint sets of colors [20, 23]. Each color used in the coloring defines what is called a stable set of the graph, i.e. a subset of pairwise nonadjacen t vertices. W e say that a graph G is k -fold x -c olor able if G admits a k -fold x -coloring. The smallest n um b er x suc h that a graph G is k -fold x -colorable is called the k -th chr omatic numb er of G and is denoted b y χ k ( G ) [23]. Ob viously , χ 1 ( G ) = χ ( G ) is the conv entional chr omatic numb er of G . This v arian t of the conv en tional graph coloring w as introduced in the context of radio frequency assignmen t problem [15, 21]. Other applications include sc heduling problems, bandwidth allo cation in radio net works, fleet main tenance and traffic phasing problems [1, 10, 13, 16]. Let n and p b e integers such that p ≥ 1 and n ≥ 2 p . As defined by T rotter, the web W n p is the graph whose v ertices can b e lab elled as { v 0 , v 1 , . . . , v n − 1 } in such a wa y that its edge set is { v i v j | p ≤ | i − j | ≤ n − p } [24]. The antiweb W n p is defined as the complemen t of W n p . Examples are depicted in Figure 1, where the v ertices are named according to an appropriate lab elling (for the sak e of conv enience, w e often name the v ertices in this w ay in the remaining of the text). W e observe that these definitions are interc hanged in some references (see [19, 25], for instance). W ebs and antiw ebs form a class of graphs that play an imp ortant role in the con text of stable sets and vertex coloring problems [3, 4, 6, 7, 9, 17, 18, 19, 25]. (a) W 8 3 (b) W 8 3 Figure 1: Example of a w eb and an an tiw eb. In this pap er, we derive a closed formula for the k -th chromatic num b er of w ebs and an tiwebs. More sp ecifically , we pro ve that χ k ( W n p ) = l kn p m and χ k ( W n p ) =  kn b n p c  , for every k ∈ N , thus generalizing similar results for o dd cycles [23]. The denominator of each of these form ulas is the size of the largest stable set in the corresponding graph, i.e. the stability numb er of the graph [24]. Besides this direct relation with the stability num b er, w e also relate the k -th chromatic num b er of webs and antiw ebs with other parameters of the graph, such as the clique, chromatic and fractional chromatic n umbers. Particularly , w e derive necessary and sufficient conditions under whic h the classical b ounds given by these parameters are tigh t. In addition to the v alue of k -th chromatic n umber, w e also pro vide optimal k -fold colorings of W n p and W n p . Based on the optimal colorings, w e analyse when webs and an tiw ebs are critical with respect to this parameter. A graph G is said to be χ -critic al if χ ( G − v ) < χ ( G ), for all v ∈ V ( G ). An immediate consequence of this definition is that if v is a v ertex of a χ -critical graph G , then there exists an optimal 1-fold coloring of G such that the color of v is not assigned to any other v ertex. Not surprisingly , χ -critical subgraphs of G play an imp ortan t role in several algorithmic approaches to vertex coloring. F or instance, they are the core of the reduction procedures of the heuristic of [12] as well as they giv e facet-inducing inequalities of vertex coloring p olytop es explored in cutting-plane metho ds [2, 11, 14]. F rom this algorithmic p oin t of view, o dd holes and odd an ti-holes are (along with cliques) the most widely used χ -critical subgraphs. It is already b een noted that not only o dd holes or o dd an ti-holes, but also χ -critical w ebs and an tiwebs giv e facet-defining inequalities [2, 18]. W e extend the concept of χ -critical graphs to χ k -critical graphs in a straightforw ard w ay . Then, w e characterize χ k -critical webs and an tiwebs, for an y integer k ≥ 1. The c haracterization crucially dep ends on the greatest common divisors b et ween n and p and b et ween n and the stabilit y n um b er (which are equal for w ebs but may b e different for an tiwebs). Using the B ´ ezout’s identit y , w e show that there exists k ≥ 1 such that W n p is χ k -critical if, and only if, gcd( n, p ) = 1. Moreo ver, when this condition holds, we determine all v alues of k for which W n p is χ k -critical. Similar results are derived for W n p , where the condition gcd( n, p ) = 1 is replaced b y gcd( n, p ) 6 = p . As a consequence, w e obtain that a w eb or an antiw eb is χ -critical if, and only if, the stability n umber divides n − 1. Such a c haracterization is trivial for w ebs but it w as still not kno wn for an tiw ebs [18]. More surprising, w e show that b eing χ -critical is also a sufficien t for a web or an an tiweb to b e χ k -critical for all k ≥ 1. Throughout this pap er, we mostly use notation and definitions consistent with what is generally accepted in graph theory . Ev en though, let us set the grounds for all the notation used from here on. Given a graph G , V ( G ) and E ( G ) stand for its set of vertices and edges, resp ectiv ely . The simplified notation V and E is prefered when the graph G is clear by the con text. The complement of G is written as G = ( V , E ). The edge defined by vertices u and v is denoted b y uv . As already mentioned, a set S ⊆ V ( G ) is said to b e a stable set if all vertices in it are pairwise non-adjacen t in G , i.e. uv 6∈ E ∀ u, v ∈ S . The stability numb er α ( G ) of G is the size of the largest stable set of G . Con v ersely , a clique of G is a subset K ⊆ V ( G ) of pairwise adjacen t vertices. The clique numb er of G is the size of the largest clique and is denoted b y ω ( G ). F or the ease of expression, w e frequently refer to the graph itself as b eing a clique (resp. stable set) if its v ertex set is a clique (resp. stable set). The fr actional chr omatic numb er of G , to b e denoted ¯ χ ( G ), is the infimum of x k among the k -fold x -colorings [22]. It is kno wn that ω ( G ) ≤ ¯ χ ( G ) ≤ χ ( G ) and n α ( G ) ≤ ¯ χ ( G ) [22]. A graph G is p erfe ct if ω ( H ) = χ ( H ), for all induced subgraph H of G . A chor d less cycle of length n is a graph G such that V = { v 1 , v 2 , . . . , v n } and E = { v i v i +1 : i = 1 , 2 , . . . , n − 1 } ∪ { v 1 v n } . A hole is a c hordless cycle of length at least four. An antihole is the complement of a hole. Holes and an tiholes are o dd or ev en according to the parity of their num b er of vertices. Odd holes and o dd an tiholes are minimally imperfect graphs [5]. Observe that the o dd holes and o dd an ti-holes are exactly the webs W 2 ` +1 ` and W 2 ` +1 2 , for some integer ` ≥ 2, whereas the cliques are exactly the webs W n 1 . In the next section, w e present general low er and upp er b ounds for the k -th chromatic n umber of an arbitrary simple graph. The exact v alue of this parameter is calculated for w ebs (Subsection 3.1) and antiw ebs (Subsection 3.2). Some consequences of this result are presen ted in the follo wing sections. In Section 4, w e relate the k -th c hromatic n um b er of webs and antiw ebs to their clique, in teger and fractional chromatic n umbers. In particular, we iden tify whic h w ebs and an tiwebs achiev e the b ounds given in Section 2 and those for which these b ounds are strict. The definitions of χ k -critical and χ ∗ -critical graphs are in tro duced in Section 5, as a natural extension of the concept of χ -critical graphs. Then, we identify all w ebs and an tiwebs that ha ve these t wo properties. 2 Bounds for the k -th c hromatic n um b er of a graph Tw o simple observ ations lead to low er and upp er b ounds for the k -th chromatic n umber of a graph G . On one hand, every v ertex of a clique of G must receiv e k colors different from any color assigned to the other vertices of the clique. On the other hand, a k -fold coloring can b e obtained b y just replicating an 1-fold coloring k times. Therefore, w e get the follo wing b ounds whic h are tight, for instance, for p erfect graphs. Lemma 1 F or every k ∈ N , ω ( G ) ≤ ¯ χ ( G ) ≤ χ k ( G ) k ≤ χ ( G ) . Another low er b ound is related to the stability n umber, as follo ws. The lexicographic pro duct of a graph G by a graph H is the graph that w e obtain by replacing each v ertex of G b y a cop y of H and adding all edges b etw een tw o copies of H if and only if the t wo replaced v ertices of G were adjacent. More formally , the lexic o gr aphic pr o duct G ◦ H is a graph suc h that: 1. the vertex set of G ◦ H is the cartesian pro duct V ( G ) × V ( H ); and 2. any tw o v ertices ( u, ˆ u ) and ( v , ˆ v ) are adjacen t in G ◦ H if and only if either u is adjacen t to v , or u = v and ˆ u is adjacen t to ˆ v As noted b y Stahl, another w a y to in terpret the k -th c hromatic num b er of a graph G is in terms of χ ( G ◦ K k ), where K k is a clique with k v ertices [23]. It is easy to see that a k -fold x -coloring of G is equiv alen t to a 1-fold coloring of G ◦ K k with x colors. Therefore, χ k ( G ) = χ ( G ◦ K k ). Using this equation w e can trivially derive the following lo w er b ound for the k -th chromatic n um b er of any graph. Lemma 2 F or every gr aph G and every k ∈ N , χ k ( G ) ≥ l kn α ( G ) m . Pro of: If H 1 and H 2 are t wo graphs, then α ( H 1 ◦ H 2 ) = α ( H 1 ) α ( H 2 ) [8]. Therefore, α ( G ◦ K k ) = α ( G ) α ( K k ) = α ( G ). W e get χ k ( G ) = χ ( G ◦ K k ) ≥ l kn α ( G ◦ K k ) m = l kn α ( G ) m .  Next we will show that the lo w er b ound given b y Lemma 2 is tight for tw o classes of graphs, namely w ebs and antiw ebs. Moreov er, some graphs in these classes also ac hieve the lo wer and upp er bounds stated b y Lemma 1. 3 The k -th c hromatic n um b er of w ebs e an tiw ebs In the remaining, let n and p b e integers suc h that p ≥ 1 and n ≥ 2 p and let ⊕ stand for addition mo dulus n , i.e. i ⊕ j = ( i + j ) mo d n for i, j ∈ Z . Let N stand for the set of natural n umbers (0 excluded). The following kno wn results will b e used later. Lemma 3 (T rotter [24]) α ( W n p ) = ω ( W n p ) = j n p k and α ( W n p ) = ω ( W n p ) = p . Lemma 4 (T rotter [24]) L et n 0 and p 0 b e inte gers such that p 0 ≥ 1 and n 0 ≥ 2 p 0 . The web W n 0 p 0 is a sub gr aph of W n p if, and only if, np 0 ≥ n 0 p and n ( p 0 − 1) ≤ n 0 ( p − 1) . 3.1 W eb W e start by defining some stable sets of W n p . F or eac h integer i ≥ 0, define the follo wing sequence of in tegers: S i = h i ⊕ 0 , i ⊕ 1 , . . . , i ⊕ ( p − 1) i (1) Lemma 5 F or every inte ger i ≥ 0 , S i indexes a maximum stable set of W n p . Pro of: By the symmetry of W n p , it suffices to consider the sequence S 0 . Let j 1 and j 2 b e in S 0 . Notice that | j 1 − j 2 | ≤ p − 1 < p . Then, v j 1 v j 2 / ∈ E ( W n p ), which prov es that S 0 indexes an indep enden t set with cardinalit y p = α ( W n p ).  Using the ab ov e lemma and the sets S i , we can no w calculate the k -th chromatic n umber of W n p . The main ideia is to build a cov er of the graph b y stable sets in which each vertex of W n p is co vered at least k times. Theorem 1 F or every k ∈ N , χ k ( W n p ) = l kn p m = l kn α ( W n p ) m . Pro of: By Lemma 2, we only ha ve to show that χ k ( W n p ) ≤ l kn p m , for an arbitrary k ∈ N . F or this purp ose, we sho w that Ξ( k ) = h S 0 , S p , . . . , S ( x − 1) p i gives a k -fold x -coloring of W n p , with x = l kn p m . W e hav e that Ξ( k ) = * 0 ⊕ 0 , 0 ⊕ 1 , . . . , 0 ⊕ p − 1 | {z } S 0 , p ⊕ 0 , . . . , p ⊕ ( p − 1) | {z } S p , . . . , ( x − 1) p ⊕ 0 , . . . , ( x − 1) p ⊕ ( p − 1) | {z } S ( x − 1) p + . Since the first elemen t of S ( ` +1) p , 0 ≤ ` < x − 1, is the last element of S `p plus 1 (mo dulus n ), we hav e that Ξ( k ) is a sequence (mo dulus n ) of integer num b ers starting at 0. Also, it has l kn p m p ≥ k n elements. Therefore, each elemen t b et ween 0 and n − 1 app ears at least k times in Ξ( k ). By Lemma 5, this means that Ξ( k ) gives a k -fold l kn p m -coloring of W n p , as desired.  3.2 An tiw eb As b efore, we pro ceed b y determining stable sets of W n p that co ver each vertex at least k times. Now, w e need to b e more judicious in the choice of the stable sets of W n p . W e start b y defining the follo wing sequences (illustrated in Figure 2): S 0 = Dl t n α ( W n p ) m : t = 0 , 1 , . . . , α ( W n p ) − 1 E (2) S i = h j ⊕ 1 : j ∈ S i − 1 i , i ∈ N = h j ⊕ i : j ∈ S 0 i , i ∈ N . W e claim that each S i indexes a maxim um stable set of W n p . This will b e sho wn with the help of the following lemmas. Lemma 6 If x, y ∈ R and x ≥ y , then b x − y c ≤ d x e − d y e ≤ d x − y e . Pro of: It is clear that x − d x e ≤ 0 and d y e − y < 1. By summing up these inequalities, w e get b x − y + d y e − d x ec ≤ 0. Therefore, b x − y c ≤ d x e − d y e . T o get the second inequalit y , recall that d x − y e + d y e ≥ d x − y + y e = d x e .  Lemma 7 F or every antiweb W n p and every inte ger k ≥ 0 , j nk α ( W n p ) k ≥ pk . Pro of: Since α ( W n p ) = j n p k , we ha v e that n p ≥ α ( W n p ), which implies nk α ( W n p ) ≥ pk . Since pk is in teger, the result follo ws.  Lemma 8 F or W n p and every inte ger ` ≥ 1 , l `n α ( W n p ) m − l ( ` − 1) n α ( W n p ) m ≥ p . Pro of: By Lemma 6, we get & `n α ( W n p ) ' − & ( ` − 1) n α ( W n p ) ' ≥ $ `n α ( W n p ) − ( ` − 1) n α ( W n p ) % = $ n α ( W n p ) % . The statemen t then follo ws from Lemma 7.  W e no w get the coun terpart of Lemma 5 for antiw ebs. Lemma 9 F or every inte ger i ≥ 0 , S i indexes a maximum stable set of W n p . Pro of: By the symmetry of an antiw eb and the definition of the S i ’s, it suffices to show the claimed result for S 0 . Let j 1 and j 2 b elong to S 0 . W e ha ve to sho w that p ≤ | j 1 − j 2 | ≤ n − p . F or the upp er b ound, note that | j 1 − j 2 | ≤ & ( α ( W n p ) − 1) n α ( W n p ) ' = & n − n α ( W n p ) ' . Lemma 7 implies that this last term is no more than d n − p e , that is, n − p . On the other hand, | j 1 − j 2 | ≥ min ` ≥ 1 & `n α ( W n p ) ' − & ( ` − 1) n α ( W n p ) '! . By Lemma 8, it follo ws that | j 1 − j 2 | ≥ p . Therefore, S 0 indexes an indep endent set of cardinalit y α ( W n p ).  The ab ov e lemma is the basis to give the expression of χ k ( W n p ). W e pro ceed by choosing an appropriate family of S i ’s and, then, w e sho w that it co v ers each vertex at least k times. W e first consider the case where k ≤ α ( W n p ). Lemma 10 L et b e given p ositive inte gers n , p , and k ≤ α ( W n p ) . The index of e ach vertex of W n p b elongs to at le ast k of the se quenc es S 0 , S 1 , . . . , S x ( k ) − 1 , wher e x ( k ) = l kn α ( W n p ) m . Pro of: Let ` ∈ { 1 , 2 , . . . , k } and t ∈ { 0 , 1 , . . . , α ( W n p ) − 1 } . Define A ( `, t ) as the sequence comprising the ( t + 1)-th elements of S 0 , S 1 , . . . , S x ( ` ) − 1 , that is, A ( `, t ) = Dl t n α ( W n p ) m ⊕ i : i = 0 , 1 , . . . , l `n α ( W n p ) m − 1 E . Since ` ≤ α ( W n p ), A ( `, t ) has l `n α ( W n p ) m distinct elements. Figure 2 illustrates these sets for W 10 3 . Let B ( `, t ) b e the subsequence of A ( `, t ) formed b y its first l ( ` + t ) n α ( W n p ) m − l tn α ( W n p ) m ≤ l `n α ( W n p ) m elemen ts (the inequalit y comes from Lemma 6). In Figure 2(b), B (1 , t ) relates to the n um b ers in blue whereas B (2 , t ) comprises the n umbers in blue and red. Notice that B ( `, t ) comprises (a) W 10 3 . S 0 S 1 S 2 S 3 S 4 S 5 S 6 S 7 A ( `, 0) 0 1 2 3 4 5 6 · · · A ( `, 1) 4 5 6 7 8 9 0 · · · A ( `, 2) 7 8 9 0 1 2 3 · · · ` = 1 ` = 2 (b) C (1) in blue, C (2) in red. Figure 2: Example of a 2-fold 7-coloring of W 10 3 . Recall that α ( W 10 3 ) = 3. consecutiv e integers (mo dulus n ), starting at l tn α ( W n p ) m ⊕ 0 and ending at l ( ` + t ) n α ( W n p ) m ⊕ ( − 1). Consequen tly , B ( `, t ) ⊆ B ( ` + 1 , t ). Let C (1 , t ) = B (1 , t ) and C ( ` + 1 , t ) = B ( ` + 1 , t ) \ B ( `, t ), for ` < k . Similarly to B ( `, t ), C ( `, t ) comprises consecutiv e in tegers (mo dulus n ), starting at l ( ` + t − 1) n α ( W n p ) m ⊕ 0 and ending at l ( ` + t ) n α ( W n p ) m ⊕ ( − 1). Observe that the first element of C ( `, t + 1) is the last element of C ( `, t ) plus 1 (mo dulus n ). Then, C ( ` ) = h C ( `, 0) , C ( `, 1) , . . . , C ( `, α ( W n p ) − 1) i is a sequence of consecutiv e in tegers (mo dulus n ) starting at the first elemen t of C ( `, 0), that is l ( ` − 1) n α ( W n p ) m ⊕ 0, and ending at the last element of C ( `, α ( W n p ) − 1), that is l ( α ( W n p )+ ` − 1) n α ( W n p ) m ⊕ ( − 1) = l ( ` − 1) n α ( W n p ) m ⊕ ( − 1) . This means that C ( ` ) ≡ h 0 , 1 , . . . , n − 1 i . Therefore, for each ` = 1 , 2 , . . . , k , C ( ` ) cov ers ev ery vertex once. Consequen tly , every vertex is co v ered k times by C (1) , C (2) , . . . , C ( k ), and so is cov ered at least k times by S 0 , S 1 , . . . , S x ( k ) − 1 .  No w we are ready to pro ve our main result for an tiwebs. Theorem 2 F or every k ∈ N , χ k ( W n p ) = l kn α ( W n p ) m . Pro of: By Lemma 2, we only need to sho w the inequalit y χ k ( W n p ) ≤ l kn α ( W n p ) m . Let us write k = `α ( W n p ) + i , for in tegers ` ≥ 0 and 0 ≤ i < α ( W n p ). By lemmas 9 and 10, it is straigh tforward that the stable sets S 0 , S 1 , . . . , S x − 1 , where x = l in α ( W n p ) m , induce an i -fold x -coloring of W n p . The same lemmas also giv e an α ( W n p )-fold n -coloring via sets S 0 , . . . , S n − 1 . One copy of the first coloring together with ` copies of the second one yield a k -fold coloring with `n + l in α ( W n p ) m = l kn α ( W n p ) m colors.  4 Relation with other parameters The strict relationship b et w een χ k ( G ) and α ( G ) established for webs (Theorem 1) and an ti- w ebs (Theorem 2) naturally motiv ates a similar question with resp ect to other parameters of G known to b e related to the c hromatic num b er. Particularly , we determine in this section when the b ounds presen ted in Lemma 1 are tight or strict. Prop osition 1 L et G b e W n p or W n p and k ∈ N . Then, χ k ( G ) = k χ ( G ) if, and only if, gcd( n, α ( G )) = α ( G ) or k < α ( G ) α ( G ) − r , wher e r = n mo d α ( G ) . Pro of: By theorems 1 and 2, χ k ( G ) = k χ ( G ) if, and only if, l kn α ( G ) m = k l n α ( G ) m , whic h is also equiv alen t to l kr α ( G ) m = k l r α ( G ) m . This equality trivially holds if r = 0, that is, gcd( n, α ( G )) = α ( G ). In the complementary case, l r α ( G ) m = 1 and, consequently , the equality is equiv alen t to kr α ( G ) > k − 1 or still k < α ( G ) α ( G ) − r .  Prop osition 2 L et G b e W n p or W n p and k ∈ N . Then, χ k ( G ) = k ω ( G ) if, and only if, gcd( n, p ) = p . Pro of: Let s = n mo d p . Using Lemma 3, note that n = b n/p c p + s = ω ( G ) α ( G ) + s . By theorems 1 and 2, we get χ k ( G ) =  k n α ( G )  = k ω ( G ) +  k s α ( G )  . The result then follows from the fact that s = 0 if, and only if, gcd( n, p ) = p .  As we can infer from Lemma 3, if p divides n , then so do es α ( W n p ) and α ( W n p ). Under suc h a condition, which holds for all p erfect and some non-p erfect w ebs and an tiwebs, the lo wer and upp er bounds giv en in Lemma 1 are equal. Corollary 1 L et G b e W n p or W n p and k ∈ N . Then, k ω ( G ) = χ k ( G ) = kχ ( G ) if, and only if, gcd( n, p ) = p . On the other hand, the same b ounds are alw ays strict for some webs and an tiwebs, including the minimally imp erfect graphs. Corollary 2 L et G b e W n p or W n p . If gcd( n − 1 , α ( G )) = α ( G ) and α ( G ) > 1 , then χ k ( G ) < k χ ( G ) , for al l k > 1 . Mor e over, if gcd( n − 1 , p ) = p and p > 1 , then k ω ( G ) < χ k ( G ) < k χ ( G ) , for al l k > 1 . Pro of: Assume that gcd( n − 1 , α ( G )) = α ( G ) and α ( G ) ≥ 2. Then, r := n mo d α ( G ) = 1 and α ( G ) α ( G ) − r ≤ 2. By Prop osition 1, χ k ( G ) < k χ ( G ) for all k > 1. T o sho w the other inequalit y , assume that gcd( n − 1 , p ) = p and p > 1. Then, gcd( n, p ) 6 = p . Moreo ver, α ( W n p ) = p > 1 and α ( W n p ) = n − 1 p > 1 so that gcd( n − 1 , α ( G )) = α ( G ) > 1. By the first part of this corollary and Prop osition 2, the result follows.  T o conclude this section, we relate the fractional chromatic num b er and the k -th chro- matic n umber. By definition, for any graph G , these parameters are connected as follo ws: ¯ χ ( G ) = inf  χ k ( G ) k | k ∈ N  . By theorems 1 and 2, χ k ( G ) k ≥ n α ( G ) , for every k ∈ N , and this b ound is attained with k = α ( G ). This leads to Prop osition 3 If G is W n p or W n p , then ¯ χ ( G ) = n α ( G ) . Actually , the ab ov e expression holds for a larger class of graphs, namely vertex transitiv e graphs [22]. The following property readily follo ws in the case of w ebs and an tiw ebs. Prop osition 4 L et G b e W n p or W n p and k ∈ N . Then, χ k ( G ) = k ¯ χ ( G ) if, and only if, k gcd( n,α ( G )) α ( G ) ∈ Z . Pro of: Let α = α ( G ) and g = gcd( n, α ). By theorems 1 and 2 and Prop osition 3, χ k ( G ) = k ¯ χ ( G ) if, and only if, kn α ∈ Z . Since n/g and α/g are coprimes, kn α = k ( n/g ) α/g is in teger if, and only if, k α/g ∈ Z .  By the ab ov e prop osition, giv en any w eb or antiw eb G suc h that α ( G ) do es not divide n , there are alwa ys v alues of k such that χ k ( G ) = k ¯ χ ( G ) and v alues of k such that χ k ( G ) > k ¯ χ ( G ). 5 χ k -critical w eb and an tiw ebs W e define a χ k -critic al graph as a graph G such that χ k ( G − v ) < χ k ( G ), for all v ∈ V ( G ). If this relation holds for every k ∈ N , then G is said to b e χ ∗ -critic al . No w we in vestigate these prop erties for w ebs and antiw ebs. The analysis is trivial in the case where p = 1 b ecause W n 1 is a clique. F or the case where p > 1, the following property will b e useful. Lemma 11 If G is W n p or W n p and p > 1 , then α ( G − v ) = α ( G ) and ω ( G − v ) = ω ( G ) , for al l v ∈ V ( G ) . Pro of: Let v ∈ V ( G ). Since p > 1, v is adjacen t to some vertex u . Lemmas 5 and 9 imply that there is a maxim um stable set of G con taining u . It follo ws that α ( G − v ) = α ( G ). Then, the other equality is a consequence of α ( G ) = ω ( G ).  Additionally , the greatest common divisor b etw een n and α ( G ) plays an imp ortan t role in our analysis. F or arbitrary nonzero in tegers a and b , the B ´ ezout’s iden tity guaran tees that the equation ax + by = gcd( a, b ) has an infinit y n umber of integer solutions ( x, y ). As there alw ays exist solutions with p ositiv e x , we can define t ( a, b ) = min  t ∈ N : at − gcd( a, b ) b ∈ Z  . F or our purp oses, it is sufficien t to consider a and b as p ositive in tegers. Lemma 12 L et a, b ∈ N . If gcd( a, b ) = b , then t ( a, b ) = 1 . Otherwise, 0 < t ( a, b ) < b gcd( a,b ) . Pro of: If gcd( a, b ) = b , then we clearly hav e t ( a, b ) = 1. No w, assume that gcd( a, b ) 6 = b . Define the coprime integers a 0 = a/ gcd( a, b ) and b 0 = b/ gcd( a, b ) > 1. W e hav e that t ( a, b ) = t ( a 0 , b 0 ) because gcd( a 0 , b 0 ) = 1 and at − gcd( a,b ) b = a 0 t − 1 b 0 , for all t ∈ N . By the B´ ezout’s iden tity , there are in tegers x > 0 and y such that a 0 x + b 0 y = 1. T ak e t = x mo d b 0 , that is, t = x −  x b 0  b 0 . Therefore, 0 ≤ t < b 0 and ta 0 − 1 b 0 =  − y −  x b 0  a 0  ∈ Z . Actually , t > 0 since b 0 > 1. These prop erties of t imply that 0 < t ( a, b ) = t ( a 0 , b 0 ) ≤ t < b 0 .  5.1 W eb In this subsection, Theorem 1 is used to determine the k -c hromatic n um b er of the graph obtained b y remo ving a v ertex from W n p . F or the ease of notation, along this subsection let t ? = t ( n, p ) = t ( n, α ( W n p )). Lemma 13 F or every k ∈ N and every vertex v ∈ V ( W n p ) , χ k ( W n p − v ) =        l kn p m , if gcd( n, p ) 6 = 1 ,  kn − b k t ? c p  , if gcd( n, p ) = 1 , Pro of: Let q = gcd( n, p ). First, supp ose that q > 1. Using Lemma 4, it is easy to v erify that W n/q p/q is a sugbraph of W n p − v . By Theorem 1, w e ha v e that χ k ( W n p − v ) ≥ & n q k p q ' =  nk p  . The con verse inequalit y follows as a consequence of χ k ( W n p − v ) ≤ χ k ( W n p ). No w, assume that q = 1. Claim 1 χ k ( W n p − v ) ≤  nk − b k t ? c p  . Pro of: By the symmetry of W n p , we only need to prov e the statemen t for v = v n − 1 . Since q = 1, p divides nt ? − 1. Let us use (1) to define Ξ = h S 0 , S p , . . . , S ( nt ? − 1 p − 1 ) p i , whic h is a sequence (mo dulus n ) of in teger n umbers starting at 0 and ending at n − 2. Notice that it co vers t ? times each integer from 0 to n − 2. Using this sequence  k t ?  times, w e get a  k t ?  t ?  - fold coloring of W n p − v with nt ? − 1 p  k t ?  colors. If t ? divides k , then w e are done. Otherwise, b y Theorem 1 and the fact that W n p − v ⊆ W n p , w e can hav e an additional  k −  k t ?  t ?  -fold coloring with at most l n p  k −  k t ?  t ?  m colors. Therefore, we obtain a k -fold coloring with at most nt ? − 1 p  k t ?  + l n p  k −  k t ?  t ?  m =  nk − b k t ? c p  colors.  Claim 2 χ k ( W n p − v ) ≥ l ( nt ? − 1) k pt ? m Pro of: By Theorem 1, it suffices to show that W n 0 t ? is a web included in W n p − v , where n 0 = nt ? − 1 p ∈ Z b ecause q = 1. By Lemma 12, t ? < p implying that n 0 < n . Therefore, w e only need to show that W n 0 t ? is a subgraph of W n p . First, notice that n ≥ 2 p + 1 and so n 0 ≥ 2 t ? + t ? − 1 p ≥ 2 t ? . Thus, W n 0 t ? is indeed a web. T o show that it is a subgraph of W n p , w e apply Lemma 4. On one hand, nt ? ≥ nt ? − 1 = n 0 p . On the other hand, n ( t ? − 1) ≤ n 0 ( p − 1) if, and only if, n 0 ≤ n − 1. Therefore, the tw o conditions of Lemma 4 hold.  By claims 1 and 2, we get & nk −  k t ?  p ' ≥ χ ( W n p − v ) ≥ & nk − k t ? p ' . T o conclude the pro of, we sho w that equalit y holds ev erywhere ab ov e. Let us write k =  k t ?  t ? + r , where 0 ≤ r < t ? . By the definition of t ? , we ha ve that nt ? − 1 p ∈ Z but nr − 1 p / ∈ Z . It follo ws that & nk − k t ? p ' ≥ & nk −  k t ?  − 1 p ' = nt ? − 1 p  k t ?  +  nr − 1 p  =  nt ? − 1 p  k t ?  + nr p  = & nk −  k t ?  p ' .  Remark 1 The pr o of of L emma 13 pr ovides the alternative e quality χ k ( W n p − v ) = l kn − k t ? p m when gcd( n, p ) = 1 . Remo ving a vertex from a graph ma y decrease its k -th c hromatic num b er of a v alue v arying from 0 to k . F or w ebs, the expressions of χ k ( W n p ) and χ k ( W n p − v ) given ab o ve together with Lemma 6 b ound this decrease as follows. Corollary 3 L et k ∈ N and v ∈ V ( W n p ) . If gcd( n, p ) 6 = 1 , then χ k ( W n p ) = χ k ( W n p − v ) . Otherwise, j k pt ? k ≤ χ k ( W n p ) − χ k ( W n p − v ) ≤ l k pt ? m . Remark 2 An imp ortant fe atur e of a χ -critic al gr aph G is that, for every vertex v ∈ V ( G ) , ther e is always an optimal c oloring wher e v do es not shar e its c olor with the other vertic es. Such a pr op erty makes it e asier to show that ine qualities b ase d on χ -critic al gr aphs ar e fac et- defining for 1 -fold c oloring p olytop es [2, 14, 18]. F or k ≥ 2 , Cor ol lary 3 establishes that cliques ar e the unique webs for which ther e exists an optimal k -fold c oloring wher e a vertex do es not shar e any of its k c olors with the other vertic es. Inde e d, for p ≥ 2 and k ≥ 2 , the upp er b ound given in Cor ol lary 3 le ads to χ k ( W n p ) − χ k ( W n p − v ) ≤  k 2  < k . Next, we iden tify the v alues of n , p , and k for whic h the low er b ound given in Corollary 3 is nonzero. In other words, w e characterize the χ k -critical w ebs, for ev ery k ∈ N . Theorem 3 L et k ∈ N . If gcd( n, p ) 6 = 1 , then W n p is not χ k -critic al. Otherwise, the fol lowing assertions ar e e quivalent: (i) W n p is χ k -critic al; (ii) k ≥ pt ? or 0 < nk p − j nk p k ≤ k pt ? ; (iii) k ≥ pt ? or k = at ? + bp for some inte gers a ≥ 1 and b ≥ 0 . Pro of: The first part is an immediate consequence of Corollary 3. F or the second part, assume that gcd( n, p ) = 1, whic h means that nt ? − 1 p ∈ Z . Let r = k n mo d p , i.e. r p = kn p − j kn p k . So, assertion (ii) can b e rewritten as k ≥ pt ? or k ≥ r t ? with r > 0 . (3) On the other hand, by Theorem 1 and Remark 1, it follo ws that χ k ( W n p ) =  k n p  +  r p  and χ k ( W n p − v ) =  k n p  + & r − k t ? p ' . Therefore, W n p is χ k -critical if, and only if, l r p m > l r − k t ? p m . If r = 0, this means that l − k t ? p m ≤ − 1 or, equiv alen tly , k ≥ pt ? . If r ≥ 1, then the condition is equiv alen t to l r − k t ? p m ≤ 0 or still k ≥ r t ? . As r < p , w e can conclude that W n p is χ k -critical if, and only if, condition (3) holds. T o sho w that (3) implies assertion (iii), it suffices to sho w that k ≥ r t ? and r > 0 imply that there exist in tegers a ≥ 1 and b ≥ 0 such that k = at ? + bp . Indeed, notice that kn − r p ∈ Z . Then, knt ? − rt ? p = ( nt ? − 1) k +( k − r t ? ) p ∈ Z . W e can deduce that k − r t ? p ∈ Z or, equiv alen ty , k = r t ? + bp for some b ∈ Z . Since k ≥ r t ? and r ≥ 1, the desired result follo ws. Con versely , let us assume that k = at ? + bp for some integers a ≥ 1 and b ≥ 0. If a ≥ p , then w e trivially get condition (3). So, assume that a < p . W e claim that r = a . Indeed, r = ( nat ? ) mo d p = nat ? −  ( nt ? − 1) a + a p  p = a −  a p  p = a. Since a ≥ 1 and b ≥ 0, w e ha v e that k ≥ r t ? and r > 0.  As an immediate consequence of Theorem 3(iii), we ha ve the characterization of χ ∗ - critical w ebs. Theorem 4 The fol lowing assertions ar e e quivalent: (i) W n p is χ ∗ -critic al; (ii) W n p is χ -critic al; (iii) α ( W n p ) divides n − 1 . Pro of: Since an y χ ∗ -critical graph is χ -critical, w e only need to pro ve that (ii) implies (iii), and (iii) implies (i). Moreov er, (iii) is equiv alen ty to t ? = gcd( n, p ) = 1. T o show the first implication, we apply Theorem 3(iii) with k = 1. It follows that gcd( n, p ) = 1 and at ? ≤ 1 for a ≥ 1. Therefore, t ? = gcd( n, p ) = 1. F or the second part, notice that any k ∈ N can b e written as k = at ? + bp for a = k ≥ 1 and b = 0, whenever t ? = 1. The result follows again b y Theorem 3(iii).  Corollary 4 Cliques, o dd holes and o dd anti-holes ar e al l χ ∗ -critic al. 5.2 An tiw ebs No w, we turn our atten tion to W n p . Similarly to the previous subsection, Theorem 2 is used to determine the k -chromatic num b er of the graph obtained by removing a v ertex from W n p . In this subsection, let t ? = t ( n, α ( W n p )). Lemma 14 F or every k ∈ N and every vertex v ∈ V ( W n p ) , χ k ( W n p − v ) =      l kn α ( W n p ) m if gcd( n, p ) = p, l k ( n − 1) α ( W n p ) m if gcd( n, p ) 6 = p. Pro of: First assume that p divides n . Using Lemma 1 and Corollary 1, we get k ω ( W n p − v ) ≤ χ k ( W n p − v ) ≤ χ k ( W n p ) = k ω ( W n p ) . By Lemma 11, ω ( W n p ) = ω ( W n p − v ) if p > 1. The same equalit y trivially holds when p = 1 since W n 1 has no edges. These facts and the ab o ve expression sho w that χ k ( W n p − v ) = χ k ( W n p ) = l kn α ( W n p ) m . No w assume that gcd( n, p ) 6 = p . Then, p > 1 and n > 2 p . By lemmas 2 and 11, we ha ve that χ k ( W n p − v ) ≥ l k ( n − 1) α ( W n p ) m . Now, w e claim that W n p − v is a subgraph of W n − 1 p . First, notice that this an tiweb is w ell-defined b ecause n − 1 ≥ 2 p . Now, let v i v j ∈ E ( W n p − v ) ⊂ E ( W n p ). Then | i − j | < p or | i − j | > n − p > ( n − 1) − p . Therefore, v i v j ∈ E ( W n − 1 p ). This pro v es the claim. Then, Theorem 1 implies that χ k ( W n p − v ) ≤ χ k ( W n − 1 p ) =  k ( n − 1) α ( W n − 1 p )  . Moreov er, since p do es not divide n , it follo ws that α ( W n − 1 p ) = j n − 1 p k = j n p k = α ( W n p ). This sho ws the con verse inequalit y χ k ( W n p − v ) ≤ l k ( n − 1) α ( W n p ) m .  Using again Lemma 6, w e can now b ound the difference b et ween χ k ( W n p ) and χ k ( W n p − v ). Corollary 5 L et k ∈ N and v ∈ V ( W n p ) . If p divides n , then χ k ( W n p − v ) = χ k ( W n p ) . Otherwise, j k α ( W n p ) k ≤ χ k ( W n p ) − χ k ( W n p − v ) ≤ l k α ( W n p ) m . Remark 3 F or k ≥ 2 , no antiweb has an optimal k -fold c oloring wher e a vertex do es not shar e any of its k c olors with other vertic es. Sinc e α ( W n p ) ≥ 2 , Cor ol lary 5 establishes that χ k ( W n p ) − χ k ( W n p − v ) ≤  k 2  < k , whenever k ≥ 2 . The ab o ve results also allo w us to characterize the χ k -critical an tiwebs, as follo ws. Theorem 5 L et k ∈ N . If gcd( n, p ) = p , then W n p is not χ k -critic al. Otherwise, the fol lowing assertions ar e e quivalent: (i) W n p is χ k -critic al; (ii) k ≥ α ( W n p ) or 0 < nk α ( W n p ) − j nk α ( W n p ) k ≤ k α ( W n p ) ; (iii) k ≥ α ( W n p ) or k = at ? + bq for some inte gers a ≥ 1 and b ≥ a (gcd( n,α ( W n p )) − t ? ) q , wher e q = α ( W n p ) / gcd( n, α ( W n p )) . Pro of: W e use Theorem 2 and Lemma 14 to get the expressions of χ k ( W n p ) and χ k ( W n p − v ). Then, the first part of the statement immediately follows. Now assume that gcd( n, p ) 6 = p . Let α = α ( W n p ) and r = k n mo d α so that r α = kn α −  kn α  . It follo ws that χ k ( W n p ) =  k n α  + l r α m and χ k ( W n p − v ) =  k n α  +  r − k α  . Therefore, W n p is χ k -critical if, and only if,  r α  >  r − k α  . If r = 0, this means that  − k α  ≤ − 1 or, equiv alently , k ≥ α . If r ≥ 1, then the condition is equiv alent to  r − k α  ≤ 0 or still k ≥ r . As r < α , we can conclude that W n p is χ k -critical if, and only if, k ≥ α, or k ≥ r and r > 0 . (4) Notice that this is exactly assertion (ii). T o show the remaining equiv alence, we use again (4). Let g = gcd( n, α ). By the defini- tions of r and t ? , we hav e that g k − r t ? α = nk − r α t ? − nt ? − g α k ∈ Z . It follows that k = at ? + bq for some b ∈ Z and a = r /g ∈ Z . Therefore, the second alternative of (4) implies the second alternativ e of assertion (iii). This leads to one direction of the desired equiv alence. Con versely , let us assume that assertion (iii) holds, that is, there exist integers a ≥ 1 and b such that k = at ? + bq and bq ≥ ag − at ? . Then, k ≥ ag . If ag ≥ α , then w e trivially get item (ii). So, assume that ag < α . W e will sho w that r = ag . Indeed, r = ( nat ? + nb g α ) mo d α = ( nat ? ) mo d α = nat ? −  ( nt ? − g ) a + ag α  α = ag − j ag α k α = ag . Since a ≥ 1 and k ≥ ag , we hav e that k ≥ r and r > 0, showing the conv erse implication.  The coun terpart of Theorem 4 for antiw ebs can b e stated no w. Theorem 6 The fol lowing assertions ar e e quivalent: (i) W n p is χ ∗ -critic al; (ii) W n p is χ -critic al; (iii) α ( W n p ) divides n − 1 . Pro of: Let α = α ( W n p ), g = gcd( n, α ) and q = α/g . It is trivial that (i) implies (ii). No w assume that W n p is χ -critical. By applying Theorem 5(iii) with k = 1, we hav e that at ? + bq = 1 and bq ≥ ag − at ? , for some int egers a ≥ 1 and b . Then, ag ≤ 1. It follo ws that a = g = 1 and b = 1 − t ? q ∈ Z . Since gcd( n, p ) 6 = p , due to Theorem 5, and 1 ≤ t ? < q , due to Lemma 12, w e obtain that 0 ≥ b ≥ l 1 − q q m = 0. Therefore, t ? = g = 1 showing that α divides n − 1. Con versely , assume that n − 1 α ∈ Z , i.e. t ? = g = 1. Then, α 6 = n p , whic h implies that gcd( n, p ) 6 = p . Moreo ver, an y k ∈ N can b e written as k = at ? + bq for a = k ≥ 1 and b = 0. 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