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On the character degree graph of solvable groups


Authors: Zeinab Akhlaghi, Carlo Casolo, Silvio Dolfi, Khatoon Khedri and Emanuele Pacifici
Journal: Proc. Amer. Math. Soc. 146 (2018), 1505-1513
MSC (2010): Primary 20C15
DOI: https://doi.org/10.1090/proc/13879
Published electronically: December 4, 2017
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Abstract: Let $ G$ be a finite solvable group, and let $ \Delta (G)$ denote the prime graph built on the set of degrees of the irreducible complex characters of $ G$. A fundamental result by P. P. Pálfy asserts that the complement $ \bar {\Delta }(G)$ of the graph $ \Delta (G)$ does not contain any cycle of length $ 3$. In this paper we generalize Pálfy's result, showing that $ \bar {\Delta }(G)$ does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of $ \Delta (G)$ can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if $ n$ is the clique number of $ \Delta (G)$, then $ \Delta (G)$ has at most $ 2n$ vertices. This confirms a conjecture by Z. Akhlaghi and H. P. Tong-Viet, and provides some evidence for the famous $ \rho $-$ \sigma $ conjecture by B. Huppert.


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Additional Information

Zeinab Akhlaghi
Affiliation: Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 15914 Tehran, Iran
Email: z_akhlaghi@aut.ac.ir, zeinab_akhlaghi@yahoo.com

Carlo Casolo
Affiliation: Dipartimento di Matematica e Informatica U. Dini, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy
Email: carlo.casolo@unifi.it

Silvio Dolfi
Affiliation: Dipartimento di Matematica e Informatica U. Dini, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy
Email: dolfi@math.unifi.it

Khatoon Khedri
Affiliation: Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111 Isfahan, Iran
Email: k.khedri@math.iut.ac.ir, khatoon_khedri@yahoo.com

Emanuele Pacifici
Affiliation: Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy
Email: emanuele.pacifici@unimi.it

DOI: https://doi.org/10.1090/proc/13879
Received by editor(s): July 21, 2016
Received by editor(s) in revised form: July 22, 2016, and June 14, 2017
Published electronically: December 4, 2017
Additional Notes: The second, third and fifth author were partially supported by the Italian INdAM-GNSAGA
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2017 American Mathematical Society

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