[Paper Review] Maximum number of colourings. II. 5-chromatic graphs

Fiachra Knox, Bojan Mohar|arXiv (Cornell University)|Oct 18, 2017

Limits and Structures in Graph Theory20 references3 citations

TL;DR

This paper proves a strengthened version of Tomescu's 1971 conjecture for 5-chromatic graphs, showing that the maximum number of $k$-colourings is achieved uniquely by graphs formed from $K_5$ by attaching trees to each vertex. The proof uses structural decomposition, chromatic polynomial analysis via the shifted polynomial $Q_G(y)$, and extensive computer-assisted case analysis on critical graphs, confirming the bound $Q_G(y) \leq (y+1)y^{n-4}(y-1)(y-2)(y-3)$ for $y \geq 4$. The result extends prior work on 4-chromatic graphs and sets the stage for a general proof.

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 $χ(G) = k \geq 4$ 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 $k=5$.

Motivation & Objective

• To resolve Tomescu's 1971 conjecture on the maximum number of $k$-colourings in connected $k$-chromatic graphs for $k=5$.
• To extend the method used in prior work on 4-chromatic graphs to the 5-chromatic case using structural decomposition and chromatic polynomial bounds.
• To establish that the extremal graphs achieving the upper bound are precisely those formed by attaching trees to each vertex of $K_5$, with equality holding for all real $y$.

Proposed method

• Uses the shifted chromatic polynomial $Q_G(y) = P_G(y+1)$ to analyze $k$-colourings in terms of $y = x-1$, simplifying asymptotic comparisons.
• Applies a recursive inequality from Lemma 2.1 to bound $Q_G(y)$ based on vertex neighbourhoods and partitions into independent sets.
• Employs case analysis on 5-critical graphs, leveraging known lists of small critical graphs (up to 13 vertices) and computer verification via programs like auty_ge g and Royle’s list.
• Uses structural decomposition: if $G$ has a 4-vertex independent set $S$, then $G-S$ is 4-chromatic and bounded via Theorem 1.2, enabling inductive bounds on $Q_G(y)$.
• Applies recursive bounds by successively removing vertices in $S$, updating the chromatic polynomial bound at each step using the $B(y)$-based inequality framework.
• Verifies all candidate graphs (e.g., Ramsey $(4,4)$ graphs on 13 vertices) computationally, confirming $Q_G(y) \ll (y+1)y^{n-4}(y-1)(y-2)(y-3)$.

Experimental results

Research questions

• RQ1 Does every connected 5-chromatic graph $G$ on $n$ vertices satisfy $Q_G(y) \leq (y+1)y^{n-4}(y-1)(y-2)(y-3)$ for all integers $y \geq 4$?
• RQ2 Are the only graphs achieving equality in this bound precisely those formed by attaching trees to each vertex of $K_5$?
• RQ3 Can the method used for 4-chromatic graphs be extended to prove the conjecture for $k=5$ using structural decomposition and computer-assisted verification?

Key findings

• The maximum number of $k$-colourings for any connected 5-chromatic graph $G$ on $n$ vertices is bounded above by $k!(k-1)^{n-k} = 5! \cdot 4^{n-5}$, with equality if and only if $G$ is obtained from $K_5$ by attaching trees to each vertex.
• The bound $Q_G(y) \leq (y+1)y^{n-4}(y-1)(y-2)(y-3)$ holds for all integers $y \geq 4$, and equality holds for all real $y$ precisely for the $K_5$-with-trees graphs.
• Extensive computer verification confirms that all 5-critical graphs on up to 13 vertices satisfy the strict inequality $Q_G(y) \ll (y+1)y^{n-4}(y-1)(y-2)(y-3)$, except for the extremal $K_5$-with-trees case.
• The proof relies on decomposing the graph via a 4-vertex independent set and applying recursive bounds using Lemma 2.1, with $B(y)$-based error terms that are controlled through case analysis.
• The method successfully generalizes the 4-chromatic case to $k=5$, and the authors expect it to extend to $k=6$ and beyond, forming a foundation for a general proof.