Most direct product of graphs are Type 1
share
A k-total coloring of a graph G is an assignment of k colors to its elements (vertices and edges) so that adjacent or incident elements have different colors. The total chromatic number is the smallest integer k for which the graph G has a k-total coloring. Clearly, this number is at least Δ(G)+1, where Δ(G) is the maximum degree of G. When the lower bound is reached, the graph is said to be Type 1. The upper bound of Δ(G)+2 is a central problem that has been open for fifty years, is verified for graphs with maximum degree 4 but not for regular graphs. Most classified direct product of graphs are Type 1. The particular cases of the direct product of cycle graphs C_m × C_n, for m =3p, 5ℓ and 8ℓ with p ≥ 2 and ℓ≥ 1, and arbitrary n ≥ 3, were previously known to be Type 1 and motivated the conjecture that, except for C_4 × C_4, all direct product of cycle graphs C_m × C_n with m,n ≥ 3 are Type 1. We give a general pattern proving that all C_m × C_n are Type 1, except for C_4 × C_4. dditionally, we investigate sufficient conditions to ensure that the direct product reaches the lower bound for the total chromatic number.
READ FULL TEXT