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Maximum number of colourings and Tomescu's conjecture

Banff International Research Station for Mathematical Innovation and Discovery

It is proved that every connected graph $G$ on $n$ vertices with $\chi(G) \geq 4$ has at most $k(k-1)^{n-3}(k-2)(k-3)$ $k$-colourings for every $k \geq 4$. Equality holds for some (and then for every) $k$ if and only if the graph is formed from $K_4$ by repeatedly adding leaves. This confirms (a strengthening of) the $4$-chromatic case of a long-standing conjecture of Tomescu [Le nombre des graphes connexes $k$-chromatiques minimaux aux sommets etiquetes, C. R. Acad. Sci. Paris 273 (1971), 1124--1126]. Proof methods may be of independent interest. In particular, one of our auxiliary results about list-chromatic polynomials solves a recent conjecture of Brown, Erey, and Li. (Joint work with Fiachra Knox.)

Mathematics; Combinatorics; Computer science; Discrete mathematics

video/mp4

Author affiliation: Simon Fraser University

2018-04-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/65267

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/