Maximum number of colourings: 4-chromatic graphs

A proof of Tomescu's graph coloring conjecture

“…Here (x) k is the falling factorial x(x − 1) · · · (x − k + 1). Knox and Mohar [21,22] proved this more general conjecture for k = 4 and 5, but our methods meet an obstruction to resolving the general case x > k. We relied on the existence of radiant vertices in Lemma 5 to recover the factor of k! when x = k, and such vertices do not necessarily exist in x-colorings of graphs of chromatic number k when x > k.…”

“…What is the maximum of P G (k) over all k-critical graphs on n vertices? Knox and Mohar [21] recently announced the bound P G (k) ≤ (k − 1) n−c log n (k − 2) c log n holds for every graph G on n vertices of minimum degree at least three and with no twins (two vertices with identical neighborhoods), where c > 0 is an absolute constant. This same bound holds for every k-critical graph with k ≥ 4 because critical graphs have no twins, and improves on Theorem 1 when n is superexponentially large in k 2 .…”

“…Tomescu observed that if k = 2, inequality (1.1) holds as an equality for every connected bipartite graph, whereas it is false for k = 3 when G is an odd cycle of length at least five. This shows that the condition k ≥ 4 is necessary; see [21,37,39,40] for details. Tomescu [38] also proved his conjecture for 4-chromatic planar graphs.…”

“…Further partial results were obtained in [4,14,15], and the book [13] gives an overview of the area. Recently, Knox and Mohar solved the cases k = 4 [21] and k = 5 [22] of Tomescu's conjecture.…”

Maximum Number of Colourings: 5-Chromatic Case

“…Very basic questions about chromatic polynomials remain unresolved and poorly understood. We refer to part I [7] for history and motivation related to using the chromatic polynomial. In this paper we continue the work on maximizing the number of colourings among all connected graphs of given order, with the goal to prove a conjecture of Tomescu [10] dating back to 1971.…”