An incidence in a graph G is a pair (v, e) with v ∈ V (G) and e ∈ E(G), such that v and e are incident. Two incidences (v, e) and (w, f) are adjacent if v = w, or e = f, or the edge vw equals e or f. The incidence chromatic number of G is the smallest k for which there exists a mapping from the set of incidences of G to a set of k colors that assigns distinct colors to adjacent incidences. In this paper, we prove that the incidence chromatic number of the toroidal grid Tm,n = Cm2Cn equals 5 when m, n ≡ 0(mod 5) and 6 otherwise.
@article{bwmeta1.element.doi-10_7151_dmgt_1663,
author = {\'Eric Sopena and Jiaojiao Wu},
title = {The Incidence Chromatic Number of Toroidal Grids},
journal = {Discussiones Mathematicae Graph Theory},
volume = {33},
year = {2013},
pages = {315-327},
zbl = {1304.05053},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1663}
}
Éric Sopena; Jiaojiao Wu. The Incidence Chromatic Number of Toroidal Grids. Discussiones Mathematicae Graph Theory, Tome 33 (2013) pp. 315-327. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1663/
[1] I. Algor and N. Alon, The star arboricity of graphs, Discrete Math. 75 (1989) 11-22. doi:10.1016/0012-365X(89)90073-3[Crossref] | Zbl 0684.05033
[2] R.A. Brualdi and J.J. Quinn Massey, Incidence and strong edge colorings of graphs, Discrete Math. 122 (1993) 51-58. doi:10.1016/0012-365X(93)90286-3[Crossref]
[3] P. Erdős and J. Nešetřil, Problem, In: Irregularities of Partitions, G. Hal´asz and V.T. S´os (Eds.) (Springer, New-York) 162-163.
[4] G. Fertin, E. Goddard and A. Raspaud, Acyclic and k-distance coloring of the grid, Inform. Proc. Lett. 87 (2003) 51-58. doi:10.1016/S0020-0190(03)00232-1[Crossref]
[5] B. Guiduli, On incidence coloring and star arboricity of graphs, Discrete Math. 163 (1997) 275-278. doi:10.1016/0012-365X(95)00342-T[Crossref] | Zbl 0871.05022
[6] M. Hosseini Dolama and E. Sopena, On the maximum average degree and the incidence chromatic number of a graph, Discrete Math. Theor. Comput. Sci. 7 (2005) 203-216. | Zbl 1153.05318
[7] M. Hosseini Dolama, E. Sopena and X. Zhu, Incidence coloring of k-degenerated graphs, Discrete Math. 283 (2004) 121-128. doi:10.1016/j.disc.2004.01.015[Crossref] | Zbl 1064.05057
[8] C.I. Huang, Y.L. Wang and S.S. Chung, The incidence coloring numbers of meshes, Comput. Math. Appl. 48 (2004) 1643-1649. doi:10.1016/j.camwa.2004.02.006[Crossref] | Zbl 1062.05059
[9] D. Li and M. Liu, Incidence coloring of the squares of some graphs, Discrete Math. 308 (2008) 6569-6574. doi:10.1016/j.disc.2007.11.047[Crossref][WoS]
[10] X. Li and J. Tu, NP-completeness of 4-incidence colorability of semi-cubic graphs, Discrete Math. 308 (2008) 1334-1340. doi:10.1016/j.disc.2007.03.076[WoS]
[11] M. Maydanskiy, The incidence coloring conjecture for graphs of maximum degree 3, Discrete Math. 292 (2005) 131-141. doi:10.1016/j.disc.2005.02.003[Crossref] | Zbl 1059.05051
[12] W.C. Shiu, P.C.B. Lam and D.L. Chen, On incidence coloring for some cubic graphs, Discrete Math. 252 (2002) 259-266. doi:10.1016/S0012-365X(01)00457-5[Crossref]
[13] W.C. Shiu and P.K. Sun, Invalid proofs on incidence coloring, Discrete Math. 308 (2008) 6575-6580. doi:10.1016/j.disc.2007.11.030[Crossref][WoS] | Zbl 1191.05045
[14] E. Sopena and J. Wu, Coloring the square of the Cartesian product of two cycles, Discrete Math. 310 (2010) 2327-2333. doi:10.1016/j.disc.2010.05.011[Crossref][WoS] | Zbl 1213.05101
[15] S.D. Wang, D.L. Chen and S.C. Pang, The incidence coloring number of Halin graphs and outerplanar graphs, Discrete Math. 25 (2002) 397-405. doi:10.1016/S0012-365X(01)00302-8[Crossref]
[16] J. Wu, Some results on the incidence coloring number of a graph, Discrete Math. 309 (2009) 3866-3870. doi:10.1016/j.disc.2008.10.027[Crossref][WoS] | Zbl 1250.05051