A proof of Tomescu's graph coloring conjecture

Abstract

In 1971, Tomescu conjectured that every connected graph G on n vertices with chromatic number k ≥ 4 has at most k!(k − 1) n−k proper k-colorings. Recently, Knox and Mohar proved Tomescu's conjecture for k = 4 and k = 5. In this paper, we complete the proof of Tomescu's conjecture for all k ≥ 4, and show that equality occurs if and only if G is a k-clique with trees attached to each vertex.

Maximum Number of Colourings: 5-Chromatic Case

“…After submitting this paper, Fox, He, and Manners managed to prove the basic case of Tomescu Conjecture [6] for arbitrary k when the number of colours is equal to k. Their proof uses some of our results as a basis.…”

Independent Sets in n-vertex k-chromatic, \ell-connected graphs

“…Finally, a graph G is ℓ-connected if |V (G)| > ℓ and any graph obtained by deleting fewer than ℓ vertices is connected. Recently Fox, He, and Manners [7] proved an old conjecture of Tomescu by finding the n-vertex k-chromatic connected graph with the maximum number of proper vertex colorings that uses k colors. This focus of this note is on maximizing i(G) and i t (G) within the family of n-vertex k-chromatic ℓ-connected graphs.…”

Extremal Colorings and Independent Sets

“…After the completion of our paper, several authors settled some cases of Conjecture 1 independently. Knox and Mohar proved it for = 4,5 in [ 19,20] and Fox, He, and Manners proved it for k≥4 when x=k in [ 14].…”