Equitable critical graphs

Abstract

A proper vertex coloring of a graph [Formula: see text] is equitable if the sizes of any two color classes differ by at most one. The equitable chromatic number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum [Formula: see text] such that [Formula: see text] is equitably [Formula: see text]-colorable. In this paper, we discuss some basic properties of equitable critical graphs as well as equitable [Formula: see text]-critical graphs. A graph [Formula: see text] is called equitable critical if [Formula: see text] for every proper subgraph [Formula: see text] of [Formula: see text]. [Formula: see text] is called equitable [Formula: see text]-critical if it is equitable [Formula: see text]-chromatic and equitable critical. Furthermore, we discuss that equitable vertex (edge) critical, equitable critical vertex (edge) graphs.