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 -