Disconnected Character graphs and odd Dominating sets

Abstract

Suppose $\Gamma$ is a finite simple graph. If $D$ is a dominating set of $\Gamma$ such that each $x\in D$ is contained in the set of vertices of an odd cycle of $\Gamma$, then we say that $D$ is an odd dominating set for $\Gamma$. For a finite group $G$, let $\Delta (G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In this paper, we show that the complement of $\Delta (G)$ contains an odd dominating set, if and only if $\Delta (G)$ is a disconnected graph with non-bipartite complement.

arXiv admin note: text overlap with arXiv:2002.01353, arXiv:1909.01180, arXiv:1909.03062