On Color Critical Graphs with Large Adaptable Chromatic Numbers

Abstract

When the vertices and edges are colored with k colors, an edge is called monochromatic if the edge and the two vertices incident with it all have the same color. The adapted chromatic number of a graph G,  $\chi _{a}\left( G\right) ,$ is the least integer k such that for each k-edge coloring of G the vertices of G can be colored with the same set of colors without creating any monochromatic edges. It is easy to see that $\chi _{a}\left( G\right) \le \chi \left( G\right)$. While this bound is tight, all the known graphs attaining this bound are not color critical. It is known that if G is a critical graph, then $\chi _{a}\left( G\right) \le \chi \left( G\right) -1$. In this article we construct a family of k-critical graphs whose adapted chromatic number is exactly one less than their chromatic number. This answers a question in Molloy and Thron (J Graph Theory 71:331–351, 2012). We also study the properties of graphs that are critical with respect to adaptable chromatic number.