## On the character degree graph of solvable groups

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- by Zeinab Akhlaghi, Carlo Casolo, Silvio Dolfi, Khatoon Khedri and Emanuele Pacifici PDF
- Proc. Amer. Math. Soc.
**146**(2018), 1505-1513 Request permission

## 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.

## References

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## Additional Information

**Zeinab Akhlaghi**- Affiliation: Faculty of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 15914 Tehran, Iran
- MR Author ID: 864104
- 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
- MR Author ID: 214592
- 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
- MR Author ID: 314262
- ORCID: 0000-0002-0513-4249
- 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
- MR Author ID: 730745
- ORCID: 0000-0001-8159-5584
- Email: emanuele.pacifici@unimi.it
- 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
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**146**(2018), 1505-1513 - MSC (2010): Primary 20C15
- DOI: https://doi.org/10.1090/proc/13879
- MathSciNet review: 3754337