Geodesic
The Incidence Chromatic Number of Toroidal Grids
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 315-327

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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 T_m,n = C_m □ C_n equals 5 when m, n ≡ 0( 5) and 6 otherwise.
Keywords: incidence coloring, Cartesian product of cycles, toroidal grid 
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Sopena, Éric; Wu, Jiaojiao. The Incidence Chromatic Number of Toroidal Grids. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 315-327. http://geodesic.mathdoc.fr/item/DMGT_2013_33_2_a5/