## Abstract

The number of proper q-colorings of a graph G, denoted by P_{G}(q), is an important graph parameter that plays fundamental role in graph theory, computational complexity theory and other related fields. We study an old problem of Linial and Wilf to find the graphs with n vertices and m edges which maximize this parameter. This problem has attracted much research interest in recent years, however little is known for general m, n, q. Using analytic and combinatorial methods, we characterize the asymptotic structure of extremal graphs for fixed edge density and q. Moreover, we disprove a conjecture of Lazebnik, which states that the Turán graph T_{s}(n) has more q-colorings than any other graph with the same number of vertices and edges. Indeed, we show that there are infinite many counterexamples in the range q=O(s^{2}/log s). On the other hand, when q is larger than some constant times s^{2}/log s, we confirm that the Turán graph T_{s}(n) asymptotically is the extremal graph achieving the maximum number of q-colorings. Furthermore, other (new and old) results on various instances of the Linial-Wilf problem are also established.

Original language | English (US) |
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Pages (from-to) | 236-275 |

Number of pages | 40 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 115 |

DOIs | |

State | Published - Nov 1 2015 |

### Bibliographical note

Funding Information:This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation .

Publisher Copyright:

© 2015 Elsevier Inc.

## Keywords

- Chromatic polynomial
- Lazebnik's conjecture
- Linial-Wilf problem
- Turán graph