Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, September 17, 2013, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Humberto Naves

The maximum number of q-colorings in graphs

We study an old problem of Linial and Wilf to find the graphs with n vertices and m edges which maximize the number of proper q-colorings of their vertices. In a breakthrough paper, Loh, Pikhurko and Sudakov reduced the problem to an optimization problem and solved it asymptotically for many ranges of parameters. We show that, for any possible choice of parameters, the optimization problem always has a solution which corresponds to either a complete multipartite graph or a graph obtained from complete multipartite graph by removing edges of two bipartite subgraphs. There are examples showing that each of the two classes of graphs is indeed optimal for certain instances of the problem. We then apply this structural result to the particular case when m corresponds to the number of edges of the Turan graph T_s(n) on n vertices with s nearly equal parts. A conjecture of Lazebnik from 1989 asserts that for any q >= s >= 2 the Turan graph T_s(n) has the maximum number of q-colorings among all graphs with the same number of vertices and edges. We disprove this conjecture by providing infinitely many counterexamples when s >= 11 and s + 8 =< q =< 2s - 3. On the positive side, we show that if q = Omega(s^2), then T_s(n) indeed achieves the maximum number of q-colorings.

Joint work with Jie Ma

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