We prove that the incidence chromatic number of the Cartesian product Cm2Cn of two cycles equals 5 when m,n ≡ 0 (mod 5) and 6 otherwise.

An incidence of G consists of a vertex and one of its incident edge in G. The incidence coloring problem is a variation of vertex coloring problem. The problem is to find the minimum number (called incidence coloring number) of colors assigned to every incidence of G so that the adjacent incidences are not assigned the same color. In this paper, we propose a linear time algorithm for incidence-...

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...

An incidence of a graph G is a pair (v, e) where v is a vertex of G and e an edge incident to v. Two incidences (v, e) and (w, f) are adjacent whenever v = w, or e = f , or vw = e or f . The incidence coloring game [S.D. Andres, The incidence game chromatic number, Discrete Appl. Math. 157 (2009), 1980–1987] is a variation of the ordinary coloring game where the two players, Alice and Bob, alte...

The incidence coloring game has been introduced in [S.D. Andres, The incidence game chromatic number, Discrete Appl. Math. 157 (2009), 1980– 1987] as a variation of the ordinary coloring game. The incidence game chromatic number ιg(G) of a graph G is the minimum number of colors for which Alice has a winning strategy when playing the incidence coloring game on G. In [C. Charpentier and É. Sopen...

For a given simple graph G = (V,E), we define an incidence as a pair (v, e), where vertex v ∈ V (G) is one of the ends of edge e ∈ E(G). Let us define a set of incidences I(G) = {(v, e) : v ∈ V (G)∧ e ∈ E(G)∧ v ∈ e}. We say that two incidences (v, e) and (w, f) are adjacent if one of the following holds: (i) v = w, e 6= f , (ii) e = f , v 6= w, (iii) e = {v, w}, f = {w, u} and v 6= u. By an inc...

An incidence of an undirected graph G is a pair (v, e) where v is a vertex of G and e an edge of G incident with v. Two incidences (v, e) and (w, f) are adjacent if one of the following holds: (i) v = w, (ii) e = f or (iii) vw = e or f . An incidence coloring of G assigns a color to each incidence of G in such a way that adjacent incidences get distinct colors. In 2005, Hosseini Dolama et al. [...