Tight bounds on odd chromatic number of some standard graph products

Abstract

An odd coloring of a graph $G$ is an assignment $f:V\left(G\right)\to \{1,2,\dots ,k\}$ of colors to the vertices of $G$ such that $f$ is a proper vertex coloring and for every non-isolated vertex $v$, there is a color that occurs an odd number of times within its open neighborhood. The minimum number of colors required by any odd coloring of $G$ is called the odd chromatic number of $G$ and is denoted by ${\chi }_o\left(G\right)$. In this paper, we give tight upper bounds on the odd chromatic number of various standard graph products and operations, including the lexicographic product, corona product, edge corona product and Mycielskian of a graph. Moreover, we give tight lower bounds on the odd chromatic number of corona product and edge corona product of graphs.