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THE CHARACTER GRAPH OF A FINITE GROUP IS PERFECT

Part of: Graph theory Representation theory of groups

Published online by Cambridge University Press:  18 November 2020

MAHDI EBRAHIMI Open the ORCID record for MAHDI EBRAHIMI [Opens in a new window]
MAHDI EBRAHIMI*
Affiliation:
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
*
e-mail: m.ebrahimi.math@ipm.ir
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Abstract

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. A perfect graph is a graph $\Gamma $ in which the chromatic number of every induced subgraph $\Delta $ of $\Gamma $ equals the clique number of $\Delta $ . We show that the character graph $\Delta (G)$ of a finite group G is always a perfect graph. We also prove that the chromatic number of the complement of $\Delta (G)$ is at most three.

Keywords

character degreecharacter graphperfect graph

MSC classification

Primary: 20C15: Ordinary representations and characters
Secondary: 05C17: Perfect graphs 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 104 , Issue 1 , August 2021 , pp. 127 - 131
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

This research was supported in part by a grant from the School of Mathematics, Institute for Research in Fundamental Sciences (IPM).

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