Even-power of Cycles With Many Vertices are Type 1 Total Colorable

Abstract

A power of cycle graph Cnk is the graph obtained from the chordless cycle Cn by adding an edge between any pair of vertices of distance at most k. Power of cycle graphs have been extensively studied in the literature, in particular with respect to coloring problems, and both vertex-coloring and edge-coloring problems have been solved in the class. The total-coloring problem, however, is still open for power of cycle graphs. A graph G is Type 1 if can be totally colored with Δ(G) + 1 colors and is Type 2 if can not be colored with Δ(G)+1 and can be colored with Δ(G)+2, with Δ(G) representing the maximum degree of a vertex in G. Although recent works from Campos and de Mello [C.N. Campos and C.P. de Mello. A Result on the Total Colouring of Powers of Cycles. Discrete Applied Mathematics, 155, (2007), 585–597.] and from Almeida et al. [S.M. Almeida, J.T. Belotti, M.M. Omai, and J.F.H. Brim. The Total Coloring of the 3rd and 4th Powers of Cycles. Presented at VI Latin American Workshop on Cliques in Graphs (2014).] point partial results for specific values of n and k, the total-coloring problem is far from being solved in the class. One remarkable conjecture from Campos and de Mello [C.N. Campos and C.P. de Mello. A Result on the Total Colouring of Powers of Cycles. Discrete Applied Mathematics, 155, (2007), 585–597.] states that Cnk, with 2 ≤ k < ⌊n/2⌋, is Type 2, if and only if n is odd and n < 3(k + 1). In particular, the conjecture would imply that, for each k ≥ 2, the number of Type 2 graphs is finite and every power of cycle graph Cnk with n ≥ 3(k + 1) would be Type 1. We address Campos and de Mello conjecture for power of cycle graphs and we prove a weaker version of the conjecture for even-power of cycle graphs: every Cnk with even k ≥ 2 and n ≥ 4k2 + 2k is Type 1. Our proof is actually stronger and shows that, for each even k < ⌊n/2⌋, the number of Type 2 power of cycle graphs Cnk is at most 2k2− k. Our results, therefore, constitute a strong evidence in favor of Campos and de Mello conjecture for power of cycle graphs.