Maximum Number of Colourings: 5-Chromatic Case

Abstract

In 1971, Tomescu conjectured [Le nombre des graphes connexes $k$-chromatiques minimaux aux sommets étiquetés, C. R. Acad. Sci. Paris 273 (1971), 1124--1126] that every connected graph $G$ on $n$ vertices with $\chi(G) = k \geq 4$ has at most $k!(k-1)^{n-k}$ $k$-colourings, where equality holds if and only if the graph is formed from $K_k$ by repeatedly adding leaves. In this note we prove (a strengthening of) the conjecture of Tomescu when $k=5$.

Maximum number of colourings: 4-chromatic graphs

“…The main message here is that the same method can be used for larger values of k. We see no difficulties of applying it for k = 6 and possibly for a few additional values. In a forthcoming paper [7] we use the results from [6] and from this paper as a basis of induction to tackle the general case.…”

“…As we show in [16], validity of ( 1) is relatively easy to establish when G has a critical subgraph whose order is much larger than χ(G). The proofs used in this paper already indicate that the most important property for bounding the number of colourings is having large minimum degree.…”

“…Although we could outline the proof of the above-mentioned result based on arguments used in this paper, the proof is deferred to [16], where we prove a more general version of this statement.…”

“…The proofs used in this paper already indicate that the most important property for bounding the number of colourings is having large minimum degree. The following strengthening of Theorem 1.2 will be derived in [16]:…”

“…In a forthcoming work [1], we prove the Tomescu conjecture for k = 5, but the proof becomes more complicated and needs a support of computer calculations. Finally, we treat the general case in [16].…”