The Incidence Chromatic Number of Toroidal Grids

The Incidence Chromatic Num b er of T oroidal Grids ´ Eric Sop ena ∗ and Jia o jiao W u † Univ. Bordeaux, LaBRI, UMR580 0, F-33400 T alence CNRS, LaBRI, UMR5 800, F-33400 T ale nce No vem b er 6, 2018 Abstract An incidence in a graph G is a p a ir ( v , e ) with v ∈ V ( G ) and e ∈ E ( G ), suc h that v and e are inciden t. Two incidences ( v , e ) and ( w, f ) are adjacen t if v = w , or e = f , or the edge v w equals e or f . The incidence chromatic num b er of G is the smallest k for whic h there exists a mapping from the set of incidences of G to a set of k colors that assigns distinct co lors to adjacent incidences. In this pap er, we prov e that the incidence c hromatic n u m b er of the toroidal grid T m,n = C m ✷ C n equals 5 wh e n m, n ≡ 0 (mo d 5) and 6 otherwise. Key words: Incidence coloring, Cartesian pro duc t of cycles, T oroidal grid. 2000 Mathematics Sub ject Classification: 05C15 1 In t ro duction Let G b e a graph with vertex set V ( G ) and edge set E ( G ). An incide nc e in G is a pair ( v , e ) with v ∈ V ( G ) a nd e ∈ E ( G ), suc h tha t v and e are ∗ E-mail: sopena@ labri.fr, corresp onding author. † E-mail: wujj0007@yaho o.com.t w. This work has been done while the a ut hor was visiting the LaBRI, suppor ted b y a p ostdo ctoral fello wship f ro m Bordeaux 1 Universit y . 1 inciden t. W e denote by I ( G ) the set of all incidences in G . Tw o incidences ( v , e ) and ( w , f ) are adjac ent if one of the f o llo wing holds: (i) v = w , (ii) e = f , (iii) the edge v w equals e or f . An incidenc e k -c oloring of G is a mapping from I ( G ) to a set of k col- ors suc h tha t adjacen t incidences ar e assigned distinct colors. The incidenc e chr omatic numb er χ i ( G ) of G is the smallest k suc h that G admits an inci- dence k -coloring . Incidence colorings w ere in tro duced b y Brualdi and Massey in [2]. In that pap er, the autho r s also conjectured that the relatio n χ i ( G ) ≤ ∆( G ) + 2 holds for ev ery gra ph G , where ∆( G ) denotes the maximu m degree of G . In [5], Guiduli disprov ed this Incidenc e C oloring C onje ctur e (ICC for short). In cidence coloring of v arious classes of graphs has b een considered in the lit t era t u re [5, 6, 7, 8, 9, 11, 12, 13, 15, 16] and the ICC conjecture w as pro v ed to hold for sev eral classes suc h as trees, complete graphs a nd complete bipartite g raphs [2], sub cubic graphs [11], K 4 -minor free graphs [7 ], graphs with maxim um a v erag e degree less than 22 9 [6], sq uare, hexagonal and honeycom b meshes [8], pow ers of paths [9], cubic Halin graphs [13], and Halin graphs with maxim um degree at least 5 [15]. The problem of determining whether a giv en graph has incidence c hromatic n umber at most k or not w as sho wn to b e NP-complete b y Li and T u [10]. Incidence colorings are related to v arious t yp es of v ertex, edge o r arc colorings. F or an y gra ph G , let H = H ( G ) b e the bipartite graph g iv en b y V ( H ) = V ( G ) ∪ E ( G ) and E ( H ) = { ( v , e ) : v ∈ V ( G ) , e ∈ E ( G ) , e and v are inciden t in G } . Eac h edge of H corresp onds to an incidence of G a nd, therefore, an y inci- dence coloring of G corresp onds to a str ong e dge c olori ng (sometimes called a distanc e-two e dge-c oloring ) of H , t ha t is a pro per color ing of the edges of H suc h tha t each color class is an induced matc hing in H [3 ]. The s u b div ision S ( G ) of G is the graph o bt a ine d from G b y inserting a v ertex o f degree t wo on ev ery edge of G . Any incidence coloring of G then corresp onds to a distanc e-two vertex c oloring of the line-graph L ( S ( G )) of S ( G ) , t ha t is a v ertex coloring suc h that an y t w o v ertices having the same color are at distance at least 3. 2 Let now G ∗ b e the digraph obtained fr om G b y replacing eac h edge of G b y t w o opp osite ar cs. Any inciden ce ( v , e ) of G , with e = v w , can then b e asso ciated with the arc v w in G ∗ . Therefore, any incidence coloring of G corresp onds to an arc-coloring of G ∗ satisfying ( i ) an y t wo arcs having the same source vertex ( o f the fo r m u v and u w ) are a s signed distinct colors, ( ii ) an y t w o consecutiv e arcs (of the form uv and v w ) are assigned distinct colors. Hence, for ev ery color c , the subgraph of G ∗ induced b y the c -colored arcs is a forest consisting of directed stars (whose arcs are directed to w ards the cen ter). The incidence chromatic num b er of G therefor e equals the dir e cte d star-arb oricity o f G ∗ , as in tro duced b y Algor and Alo n in [1]. Let G and H b e gr aphs . The Cartesian p r o duct G ✷ H of G and H is the graph with v ertex set V ( G ) × V ( H ) where tw o ve rtices ( u 1 , v 1 ) and ( u 2 , v 2 ) are adjacen t if a nd only if either u 1 = u 2 and v 1 v 2 ∈ E ( H ), or v 1 = v 2 and u 1 u 2 ∈ E ( G ). Let P n and C n denote resp ectiv ely the pa t h and the cycle on n vertice s. W e will denote b y G m,n = P m ✷ P n the grid with m ro ws and n columns and by T m,n = C m ✷ C n the t or oid al grid with m ro ws and n columns. In this pap er, w e determine the incidenc e c hromatic n um b er of toroidal grids a nd prov e that this class of graphs satisfies the ICC: Theorem 1 F or every m, n ≥ 3 , χ i ( T m,n ) = 5 if m, n ≡ 0 (mo d 5) and χ i ( T m,n ) = 6 otherwise. In [8], Huang, W a ng and Ch ung pro v ed that χ i ( G m,n ) = 5 for ev ery m , n . Since every toroidal grid T m,n con tains the grid G m,n as a subgraph, w e get that χ i ( T m,n ) ≥ 5 fo r ev ery m , n . The pap er is organized as follows. In Section 2 w e g ive basic prop e rties and illustrate the tec hniques w e shall use in the pro of of our main result, whic h is giv en in Section 3. 2 Preliminaries Let G b e a graph, u a ve rtex of G with maxim um degree and v a neigh b our of u . Since in any incidence coloring of G all the incidences of the form ( u, e ) ha v e to get distinct colors and all of them hav e to get a color diff erent fro m the colo r of ( v , v u ), we hav e: Prop osition 2 F or every gr aph G , χ i ( G ) ≥ ∆( G ) + 1 . 3 The squar e G 2 of a gr aph G is g iv en by V ( G 2 ) = V ( G ) and uv ∈ E ( G 2 ) if and only if u v ∈ E ( G ) o r there exists w ∈ V ( G ) suc h that u w , v w ∈ E ( G ). In other w ords, any t w o v ertices within distance at most tw o in G are linke d b y an edge in G 2 . Let no w c b e a prop er v ertex coloring o f G 2 and µ b e the mapping defined by µ ( u, uv ) = c ( v ) for ev ery inciden ce ( u, uv ) in I ( G ). It is not difficult to c hec k that µ is indeed an incidence coloring of G (see Example 8 b elo w). Therefore w e ha v e: Prop osition 3 F or every gr aph G , χ i ( G ) ≤ χ ( G 2 ) . In [4], F ertin, Go ddard and Raspaud pro v ed that the ch romatic num b er of the square of any d -dimensional grid G n 1 ,...,n d is at most 2 d + 1, whic h thus implies the a b ov e mentioned result concerning 2-dimensional grids [8]. In [14], w e studied the c hromatic n um b er of the squares of toroidal grids and prov ed the follo wing: Theorem 4 L et T m,n = C m ✷ C n . Then χ ( T 2 m,n ) ≤ 7 exc ept that χ ( T 2 3 , 3 ) = 9 and χ ( T 2 3 , 5 ) = χ ( T 2 4 , 4 ) = 8 . By Prop osition 3, this result provid es upp e r b ounds on the incidence c hromatic n um b er of toro ida l grids. In [1 4], we also prov ed the following: Theorem 5 F or every m, n ≥ 3 , χ ( T 2 m,n ) ≥ 5 . Mor e over, χ ( T 2 m,n ) = 5 if and only if m, n ≡ 0 (mo d 5) . In [1 6], the second author prov ed t he fo llowing: Theorem 6 F or a r e gular gr aph G , χ i ( G ) = ∆( G ) + 1 if and only if χ ( G 2 ) = ∆( G ) + 1 . Since toroidal grids are 4-regular, by com bining Prop osition 2, Theorems 5 and 6 we g e t the following: Corollary 7 F or every m, n ≥ 3 , χ i ( T m,n ) ≥ 5 . Mor e over, χ i ( T m,n ) = 5 if and only if m, n ≡ 0 (mo d 5) . Note here that t his corolla r y is part of our main result. An y v ertex coloring of the square of a toroidal grid T m,n can b e giv en as an m × n matrix whose entrie s corresp ond in an o b vious w a y to the color s of the vertice s. Suc h a matrix will b e called an m × n p attern in the follo wing. 4 A = 1 2 3 4 3 4 5 6 5 6 7 8 7 8 1 2 7 8 1 2 4 2 1 3 2 4 3 1 3 4 5 6 1 2 3 4 6 4 3 5 4 6 5 3 5 6 7 8 3 4 5 6 8 6 5 7 6 8 7 5 7 8 1 2 5 6 7 8 2 8 7 1 8 2 1 7 1 2 3 4 Figure 1: A pattern A and the corresp onding incidence coloring of T 4 , 4 . 5 Example 8 Fig. 1 shows a 4 × 4 pattern A , whic h defines a v ertex coloring of T 2 4 , 4 , and the incidence coloring of T 4 , 4 induced by this pattern, according to the discussion b efore Prop osition 3. Note for instance t ha t the four inc idences of the form ( u , uv ), for u b eing the second v ertex in the third ro w, hav e color 6, whic h corresp onds to the entry in row 3, column 2, o f pattern A . If A and B are patterns o f size m × n and m × n ′ resp e ctiv ely , w e shall denote b y A + B the pattern of size m × ( n + n ′ ) obt a ine d b y “gluing” together the patterns A and B . Moreov er, w e shall denote b y ℓA , ℓ ≥ 2, the pattern of size m × ℓn obtained b y gluing together ℓ copies of the pa ttern A . W e no w shortly describe the tec hnique w e shall use in the next sec tion. The main idea is to use a pattern for coloring the square of a toroidal grid in order to get an incidence color ing of this toroidal grid. Ho w ev er, as sho wn in [1 4 ], the squares of t o roidal grids are not all 6-colora ble . Therefore, w e s hall use the notion o f a q uasi-p attern whic h corresp onds to a v ertex 6- colo ring o f the square of a sub gr aph of a toroidal grid o btained b y deleting some edges (namely t ho se edges that cause a conflict when transforming a v ertex coloring to its corresp o nding incidence coloring). W e can then use suc h a quasi- pattern in the same wa y as b efore to obtain a p artial incidence coloring of the toroidal grid. F inally , w e shall pro v e that suc h a partial incidenc e colo r ing can b e extended to the whole toroidal grid without using an y a dditional color (most of the time, sev eral distinct extensions ar e av a ila ble and we shall prop ose one of them). W e shall also use the follow ing: Remark 9 F or every m, n ≥ 3 , p, q ≥ 1 , if χ i ( T m,n ) ≤ k then χ i ( T pm,q n ) ≤ k . T o see that, it is enough to observ e that eve ry incidence k - coloring c of T m,n can b e extended to an incidence k -color ing c p,q of T pm,q n b y “ rep eating” the pat tern giv en by c , p times “v ertically” and q times “horizon tally”. 3 Pro of o f The o rem 1 According to Corollary 7, w e only need to prov e that χ i ( T m,n ) ≤ 6 fo r ev ery m, n ≥ 3. The pro of is based on a series of lemmas, a ccording to different v alues of m and n . W e first consider the case when m ≡ 0 (mo d 3). W e hav e prov ed in [1 4 ] the f o llo wing: 6 B = 3 1 2 C = 1 4 2 5 2 5 3 6 3 6 1 4 D = 1 4 3 6 2 5 1 4 3 6 2 5 E = 2 5 3 6 1 4 B + C = 3 1 4 2 5 1 2 5 3 6 2 3 6 1 4 B + D + E = 3 1 4 3 6 2 5 1 2 5 1 4 3 6 2 3 6 2 5 1 4 Figure 2: P at terns and quasi-patt erns for L emma 11. 2 5 6 1 4 5 6 3 4 1 2 4 5 2 3 1 2 5 3 6 3 6 4 2 5 6 4 1 5 2 3 5 6 3 1 2 3 6 1 4 1 4 5 3 6 4 5 2 6 3 1 6 4 1 2 3 1 4 2 5 Figure 3: Incidence coloring for Lemma 11. Prop osition 10 I f k ≥ 1 , n ≥ 3 and n even, then χ ( T 2 3 k, n ) ≤ 6 . Here we prov e: Lemma 11 If k ≥ 1 and n ≥ 3 , then χ i ( T 3 k, n ) ≤ 6 . Pro of. If n is ev en, t he result follows from Prop ositions 3 and 10. W e thus assume t ha t n is o dd, and w e let first k = 1. W e consider three cases. 1. n = 3. W e can easily get an incidence 6-coloring by coloring the incidences o f one dimension with { 1 , 2 , 3 } and the incidences of the o ther dimension with { 4 , 5 , 6 } . 7 F = 1 2 4 1 2 4 3 5 6 3 5 6 G = 1 2 3 4 1 2 3 4 3 4 5 6 3 4 5 6 H = 2 F + 2 G = 1 2 4 1 2 4 1 2 3 4 1 2 3 4 1 2 4 1 2 4 1 2 3 4 1 2 3 4 3 5 6 3 5 6 3 4 5 6 3 4 5 6 3 5 6 3 5 6 3 4 5 6 3 4 5 6 Figure 4: Quasi-patterns fo r Lemma 13. 2. n = 4 ℓ + 1. Let B and C b e the patterns depicted in Fig. 2 and consider the quasi- pattern B + ℓC (the quasi-pattern B + C is depicted in Fig. 2). This quasi-pattern prov ides a 6-coloring of T 2 m,n if w e delete all the edges linking v ertices in the first column to ve rtices in the second column. W e can use this quasi-pattern to obtain a n incidence 6-coloring of T m,n b y mo difying six incidence colors, as show n in Fig. 3 (mo dified colors are in b oxes). 3. n = 4 ℓ + 3. Let B , D and E b e the patterns depicted in Fig. 2 and consider the quasi-pattern B + ℓD + E ( t he quasi-pattern B + D + E is depicted in Fig. 2). As in the previous case, w e can use this quasi-pattern to obtain an incidence 6-colo r ing of T m,n b y mo difying the same six incidence colors. F or k ≥ 2, the result now directly fo llo ws f rom Remark 9. W e now consider the case when m ≡ 0 (mo d 4) . F or m ≡ 0 (mo d 5), w e ha v e prov ed in [14] the follo wing: Prop osition 12 I f k ≥ 1 , n ≥ 5 and n 6 = 7 , then χ ( T 2 5 k, n ) ≤ 6 . Here we prov e: Lemma 13 If k ≥ 1 , n ≥ 3 an d ( k , n ) 6 = (1 , 5) , then χ i ( T 4 k, n ) ≤ 6 . Pro of. F or n = 5 , t he result holds b y Prop osition 12, except fo r k = 1. 8 3 5 6 4 2 1 4 2 1 x x x x x x 4 2 1 4 2 1 3 5 6 1 2 4 6 5 3 6 5 3 x x x x x x 6 5 3 6 5 3 1 2 4 3 4 5 6 4 2 1 3 2 4 3 1 x x x x x x x x 4 2 1 3 2 4 3 1 3 4 5 6 1 2 3 4 6 4 3 5 4 6 5 3 x x x x x x x x 6 4 3 5 4 6 5 3 1 2 3 4 Figure 5: P artial incidence color ing s fo r Lemma 13. 9 I = 6 1 2 3 4 5 3 4 5 6 1 2 5 6 1 2 3 4 2 3 4 5 6 1 4 5 6 1 2 3 I ′ = 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 3 4 5 6 1 2 5 6 1 2 3 4 2 3 4 5 6 1 4 5 6 1 2 3 Figure 6: P at terns for Lemma 1 4. Assume no w k = 1 and n 6 = 5 and consider the quasi-pat terns F and G depicted in Fig. 4. F rom these patt erns, w e can deriv e a partial incidence 6- coloring of T 4 , 3 and T 4 , 4 , resp ectiv ely , as sho wn in Fig. 5, where the uncolored incidences are denoted by x . It is easy to c heck that ev ery suc h incidence has o nly fo ur forbidden colors and that only incidences b elonging to a same edge hav e to b e distinct. Therefore, these partial incidence colorings can b e extended to incidence 6- colorings of T 4 , 3 and T 4 , 4 . F or n ≥ 6, w e shall use the quasi-pattern H = pF + q G where p and q satisfy n = 3 p + 4 q (recall t ha t ev ery in teger except 1,2 and 5 can b e written in this form). The quasi-patt ern H = 2 F + 2 G is depicte d in Fig. 4. As in the previous case, this quasi-pattern pro vides a partial incidence 6-color ing of T 4 ,n that can b e extended to a n incidence 6- coloring o f T 4 ,n . F or k ≥ 2, the result no w directly follows fr o m Remark 9. W e now consider the remaining cases. Lemma 14 If m, n ≥ 5 , m 6 = 6 , 8 and n 6 = 7 , then χ i ( T m,n ) ≤ 6 . Pro of. Assume m, n ≥ 5 , m 6 = 6 , 8 and n 6 = 7. By Prop osition 1 2, w e ha v e χ ( T 2 5 k, n ) ≤ 6 for n 6 = 7. Hence, there exists a v ertex 6-colo ring o f T 2 5 k, n whic h corresp onds to some pattern M o f size 5 k × n . W e claim that eac h row of pattern M can b e rep eated one or three times to get quasi-patterns that can b e extended to incidence 6 - colorings of the corresp onding toroidal grids. Let f o r instance M ′ b e the quasi-pattern obtained from M b y rep eating the first row of M three times. The quasi-pattern M ′ has th us size (5 k + 2) × n . The quasi-pattern M ′ induces a partia l incidence coloring of T 5 k + 2 ,n in whic h the only uncolored incidences are those lying on the edges linking ve rtices in the first row to vertice s in the second ro w and on the edges linking vertic es in the second row t o v ertices in t he third row. 10 4 5 6 1 2 3 5 1 6 2 1 3 2 4 3 5 4 6 x x x x x x y y y y y y 5 1 6 2 1 3 2 4 3 5 4 6 z z z z z z x x x x x x 5 1 6 2 1 3 2 4 3 5 4 6 3 4 5 6 1 2 6 1 2 3 4 5 2 4 3 5 4 6 5 1 6 2 1 3 5 6 1 2 3 4 3 4 5 6 1 2 4 6 5 1 6 2 1 3 2 4 3 5 2 3 4 5 6 1 5 6 1 2 3 4 1 3 2 4 3 5 4 6 5 1 6 2 4 5 6 1 2 3 2 3 4 5 6 1 3 5 4 6 5 1 6 2 1 3 2 4 6 1 2 3 4 5 Figure 7: A partial incidence coloring of T 7 , 6 . 11 W e illustrate this in Fig. 6 with a patt ern I of size 5 × 6 (this pattern induces a ve rtex 6-coloring of T 2 5 , 6 ) and its asso ciated pattern I ′ of size 7 × 6 . The partia l incidence coloring of T 7 , 6 obtained from I ′ is then giv en in F ig. 7 , where uncolored incidences ar e denoted by x , y and z . Observ e now that in eac h column, the tw o incidences denoted by x hav e three forbidden colors in common and eac h of them has four forbidden colors in total. Therefore, we can a ssign them the same colo r. No w, in each column, the incidences denoted b y y a nd z hav e four forbidden colors in common (the color assigned to x is one of them) and eac h of them has fiv e forbidden colors in to tal. They can b e th us colored with distinct colors. Doing that, w e extend the partial incidence coloring o f T 7 , 6 to an incidence 6 -coloring of T 7 , 6 . The same tec hnique can b e used for obtaining an incidence 6-colo ring of T 5 k + 2 ,n since all the columns are “indep enden t” in the quasi-pattern M ′ , with resp ect to uncolored incidences. If w e rep eat three times sev eral distinct rows of pattern M , eac h rep eat ed ro w will pro duce a chain of four uncolored incidences, as b efore, and any t w o suc h c hains in the same column will b e “ indep enden t ” , since they will b e separated b y a n edge whose inciden ces are b oth colored. Hence, w e will b e able to extend the corresp onding quasi-pattern to an incidence 6-coloring of the t oroidal grid, b y a ssigning av aila ble colo rs to each chain as w e did a b ov e. Starting from a pa t t ern M of size 5 k × n , w e can thus obtain quasi- patterns of size (5 k + 2) × n , (5 k + 4) × n , (5 k + 6) × n and (5 k + 8) × n , b y rep eating resp ectiv ely o ne, t w o, t hree or fo ur lines from M . Using these quasi-patterns, w e can pro duce incidence 6-colorings of the toroidal grid T m,n , m, n ≥ 5, n 6 = 7, for ev ery m except m = 6, 8. The only remaining cases a r e m = 4, n = 5 and m = n = 7. Then w e ha v e: Lemma 15 χ i ( T 4 , 5 ) ≤ 6 and χ i ( T 7 , 7 ) ≤ 6 . Pro of. Let m = 4 and n = 5. Consider the pattern C of size 3 × 4 depicted in Fig. 2. As in the pro of of Lemma 14, w e can rep eat the first row of C three times to get a quasi-pattern C ′ that can b e extended to an incidenc e 6-coloring of T 5 , 4 . W e then exc hange m and n to get an incidence 6 - coloring of T 4 , 5 , depicted in Fig. 8 (the colo r s assigned to uncolored incidences are dra wn in b ox es). Let now m = n = 7 and consider the quasi-pa t t ern J depicted in Fig. 9 . This quasi-pattern pro vides the partial incidence coloring of T 7 , 7 giv en in 12 1 1 1 2 3 4 3 6 4 3 6 5 4 6 5 2 2 2 3 1 5 5 5 6 4 1 6 3 1 6 3 2 1 3 2 4 4 4 5 6 2 2 2 3 1 6 3 5 6 3 5 4 6 5 4 1 1 1 2 3 4 4 4 5 6 3 6 2 3 6 2 1 3 2 1 5 5 5 6 4 Figure 8: An incidence 6 - coloring of T 4 , 5 . J = 3 5 6 3 4 5 6 1 2 4 1 2 3 4 1 2 4 1 2 3 4 3 5 6 3 4 5 6 3 5 6 3 4 5 6 1 2 4 1 2 3 4 1 2 4 1 2 3 4 Figure 9: A quasi-pattern fo r Lemma 15. 13 1 2 4 1 2 3 4 6 4 3 1 5 2 6 4 3 5 4 6 5 3 5 6 3 5 6 1 2 3 5 6 3 4 5 6 4 2 1 4 2 1 4 2 1 3 2 4 3 1 y y y y y y y 5 6 3 5 6 1 2 4 2 1 4 2 1 4 2 1 3 2 4 3 1 3 5 6 3 4 5 6 1 2 4 1 2 3 4 6 5 3 6 5 3 6 4 3 5 4 6 5 3 x x x x x x x x x x x x x x 6 5 3 6 5 3 6 4 3 5 4 6 5 3 1 2 4 1 2 3 4 3 5 6 3 4 5 6 4 2 1 4 2 1 4 2 1 3 2 4 3 1 x x x x x x x x x x x x x x 4 2 1 4 2 1 4 2 1 3 2 4 3 1 3 5 6 3 4 5 6 Figure 10: A partial incidence 6-color ing of T 7 , 7 . 14 Fig. 1 0, where incidences with mo dified colors are in b o xes and uncolored incidences are denoted by x and y . Observ e now tha t the incidences denoted b y y ha v e fiv e for bidden colors while the incidences denoted by x ha v e four forbidden colors. Therefore, this partial coloring can b e extende d to an incidence 6-coloring o f T 7 , 7 . 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