Abstract

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 $\Delta (G)$ 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.

中文翻译：

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断开的字符图和奇数支配集

摘要

假设 Γ 是一个有限的简单图。如果D是 Γ 的支配集，使得每个$X\in D$包含在 Γ 的奇周期的顶点集中，那么我们说D是 Γ 的奇支配集。对于有限群G，令$\Delta (G)$表示建立在G的不可约复字符的度集上的字符图。在本文中，我们证明了$\Delta (G)$ 包含奇支配集，当且仅当 $\Delta (G)$ 是具有非二部补集的断开图。