On the conformable colorings of k-regular graphs

In 1988, Chetwynd and Hilton defined conformable vertex colorings when trying to characterize the vertex colorings induced by a (∆ + 1)-total coloring. Anticonformable colorings were used to characterize the subcubic conformable graphs. A graph G is anticonformable if it has a (∆ + 1)-vertex coloring such that the number of color classes (including empty color classes) with the same parity as |V| is at most def(G) = ∑v∈V (∆− dG(v)). The only connected subcubic not anticonformable graph is the triangular prism graph L3. In this paper, we prove that if k is even, then every k-regular graph is not anticonformable; and if k ≥ 3 is odd, then there is a not anticonformable graph Hk, where H3 = L3.