A polynomial-time algorithm for conformable coloring on regular bipartite and subcubic graphs

Abstract

In 1988, Chetwynd and Hilton observed that a $(\Delta +1)$-total coloring induces a vertex coloring in the graph, they called it conformable. A $(\Delta +1)$-vertex coloring of a graph $G=(V,E)$ is called conformable if the number of color classes of parity different from that of $\left|V\right|$ is at most the deficiency $def\left(G\right)={\sum }_{v\in V}(\Delta -d_G\left(v\right))$ of $G$, where $d_G\left(v\right)$ is the degree of a vertex $v$ of $V$. In 1994, McDiarmid and Sánchez-Arroyo proved that deciding whether a graph $G$ has $(\Delta +1)$-total coloring is NP-complete even when $G$ is $k$-regular bipartite with $k\ge 3$. However, the time-complexity of the problem of determining whether a graph admits a conformable coloring (Conformability problem) remains unknown. In this paper, we prove that Conformability problem is polynomial solvable for the class of $k$-regular bipartite and for the class of subcubic graphs.