Results about the total chromatic number and the conformability of some families of circulant graphs

Abstract

In this paper, we show that with the exception of the graph C12(1,3), which we prove to be Type 2, all members of the following three infinite families of 4-regular circulant graphs: Cn(2k,3), k≥1 and n=(8μ+6λ)k, for nonnegative integers μ and λ; C3n(1,3), for n≥3 and n≠4; and C3λp(1,p), for λ≥1 and p multiple of 3 are Type 1. The last two results are what is expected whether the Khennoufa and Togni’s conjecture is true, which states that 4-regular circulant graphs Cn(1,k) are Type 1 with 1<k<n2, but a finite number of Type 2 graphs. It is known that if a graph G is Type 1, then it is conformable. Furthermore, we introduce a relationship between the conformability problem and the equitable vertex coloring problem, by showing infinite families for which any equitable (Δ+1)-vertex coloring is conformable. In this context, we exhibit the infinite conformable graph family C(2q+1)n(d1,…,dq), n,q positive integers, containing C5n(d1,d2), which in turns, comprises one fifth of all 4-regular circulant graphs.