Disconnected Character graphs and odd Dominating sets

英文摘要：

Suppose $Γ$ is a finite simple graph. If $D$ is a dominating set of $Γ$ such that each $x\in D$ is contained in the set of vertices of an odd cycle of $Γ$, then we say that $D$ is an odd dominating set for $Γ$. For a finite group $G$, let $Δ(G)$ 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 $Δ(G)$ contains an odd dominating set, if and only if $Δ(G)$ is a disconnected graph with non-bipartite complement.