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Extremal Colorings and Independent Sets

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Abstract

We consider several extremal problems of maximizing the number of colorings and independent sets in some graph families with fixed chromatic number and order. First, we address the problem of maximizing the number of colorings in the family of connected graphs with chromatic number k and order n where \(k\ge 4\). It was conjectured that extremal graphs are those which have clique number k and size \({k\atopwithdelims ()2}+n-k\). We affirm this conjecture for 4-chromatic claw-free graphs and for all k-chromatic line graphs with \(k\ge 4\). We also reduce this extremal problem to a finite family of graphs when restricted to claw-free graphs. Secondly, we determine the maximum number of independent sets of each size in the family of n-vertex k-chromatic graphs (respectively connected n-vertex k-chromatic graphs and n-vertex k-chromatic graphs with c components). We show that the unique extremal graph is \(K_k\cup E_{n-k}\), \(K_1\vee (K_{k-1}\cup E_{n-k})\) and \((K_1 \vee (K_{k-1} \cup E_{n-k-c+1}))\cup E_{c-1}\) respectively.

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Acknowledgements

We would like to thank anonymous referees for their helpful comments. After the completion of our paper, several authors settled some cases of Conjecture 1 independently. Knox and Mohar proved it for \(k=4,5\) in [19, 20] and Fox, He, and Manners proved it for \(k\ge 4\) when \(x=k\) in [14].

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, Marquette University, Milwaukee, WI, 53201, USA

    John Engbers

  2. Department of Mathematics, Gebze Technical University, Kocaeli, Turkey

    Aysel Erey

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  1. John Engbers
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  2. Aysel Erey
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Correspondence to Aysel Erey.

Additional information

J. Engbers’s research was supported by the Simons Foundation Grant 524418.

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Engbers, J., Erey, A. Extremal Colorings and Independent Sets. Graphs and Combinatorics 34, 1347–1361 (2018). https://doi.org/10.1007/s00373-018-1951-3

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  • DOI: https://doi.org/10.1007/s00373-018-1951-3

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