2 research outputs found
A proof of Tomescu's graph coloring conjecture
In 1971, Tomescu conjectured that every connected graph on vertices
with chromatic number has at most proper
-colorings. Recently, Knox and Mohar proved Tomescu's conjecture for
and . In this paper, we complete the proof of Tomescu's conjecture for all
, and show that equality occurs if and only if is a -clique with
trees attached to each vertex.Comment: Adds a short proof of the case k=4, removing dependence on previous
wor
Maximum number of colourings. I. 4-chromatic graphs
It is proved that every connected graph on vertices with has at most -colourings for every . Equality holds for some (and then for every) if and only if the graph
is formed from by repeatedly adding leaves. This confirms (a
strengthening of) the -chromatic case of a long-standing conjecture of
Tomescu [Le nombre des graphes connexes -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