TY - JOUR

T1 - Maximizing proper colorings on graphs

AU - Ma, Jie

AU - Naves, Humberto

N1 - Publisher Copyright:
© 2015 Elsevier Inc.
Copyright:
Copyright 2015 Elsevier B.V., All rights reserved.

PY - 2015/11/1

Y1 - 2015/11/1

N2 - The number of proper q-colorings of a graph G, denoted by PG(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 Ts(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(s2/log s). On the other hand, when q is larger than some constant times s2/log s, we confirm that the Turán graph Ts(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.

AB - The number of proper q-colorings of a graph G, denoted by PG(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 Ts(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(s2/log s). On the other hand, when q is larger than some constant times s2/log s, we confirm that the Turán graph Ts(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.

KW - Chromatic polynomial

KW - Lazebnik's conjecture

KW - Linial-Wilf problem

KW - Turán graph

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U2 - 10.1016/j.jctb.2015.07.002

DO - 10.1016/j.jctb.2015.07.002

M3 - Article

AN - SCOPUS:84939262431

VL - 115

SP - 236

EP - 275

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

ER -