The Incidence Chromatic Number of Toroidal Grids
Discussiones Mathematicae Graph Theory, Tome 33 (2013) no. 2, p. 315.

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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.
Classification : 05C15, 05C76
Mots-clés : incidence coloring, Cartesian product of cycles, toroidal grid, Cartesian product 
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Éric Sopena; Jiaojiao Wu. The Incidence Chromatic Number of Toroidal Grids. Discussiones Mathematicae Graph Theory, Tome 33 (2013) no. 2, p. 315. https://geodesic-test.mathdoc.fr/item/DMGT_2013__33_2_268177/