Maximizing proper colorings on graphs

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 Tur\'{a}n 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 Tur\'{a}n 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.