Disconnected Character graphs and odd Dominating sets

Abstract

Suppose Γ is a finite simple graph. If D is a dominating set of Γ such that each x ∈ 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 ∆(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 ∆(G) contains an odd dominating set, if and only if ∆(G) is a disconnected graph with non-bipartite complement.