Discussiones Mathematicae Graph Theory 33(2) (2013) 315-327
doi: 10.7151/dmgt.1663

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The Incidence Chromatic Number
of Toroidal Grids

Éric Sopena

Univ. Bordeaux, LaBRI, UMR5800, F-33400 Talence
CNRS, LaBRI, UMR5800, F-33400 Talence

Jiaojiao Wu

Department of Applied Mathematics
National Sun Yat-sen University, Taiwan


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 = Cm[¯] Cn equals 5 when m,n ≡ 0 (mod 5) and 6 otherwise.

Keywords: incidence coloring, Cartesian product of cycles, toroidal grid

2010 Mathematics Subject Classification: 05C15.


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Received 23 October 1011
Revised 20 February 2012
Accepted 7 March 2012