Compositions, decompositions, and conformability for total coloring on power of cycle graphs

Abstract

Power of cycle graphs C n k have been extensively studied with respect to coloring problems, being both the vertex and the edge-coloring problems already solved in the class. The total coloring problem (of determining the minimum number of colors needed to color the vertices and the edges of a graph in a manner that no two adjacent elements are colored the same), however, is still open for power of cycle graphs. Actually, despite partial results for specific values of n and k , not even the well-known Total Coloring Conjecture is settled in the class. A remarkable conjecture by Campos and Mello of 2007 states that if C n k is neither a cycle nor a complete graph, then it has total chromatic number χ T = Δ + 2 if n is odd and n < 3 ( k + 1 ) , and χ T = Δ + 1 otherwise. We provide strong evidences for this conjecture: we settle a dichotomy for all power of cycle graphs with respect to conformability (a well-known necessary condition for a graph to have χ T = Δ + 1 ) and we develop a framework which may be used to prove that, for any fixed k , the number of C n k graphs with χ T ≠ Δ + 1 is finite. Moreover, we prove this finiteness for any even k and for k ∈ { 3 , 5 , 7 } . We also use our composition technique to provide a proof of Campos and Mello’s conjecture for all C n k graphs with k ∈ { 3 , 4 } .