Minimum non-chromatic-choosable graphs with given chromatic number

TL;DR

The paper determines the minimum vertex count of a $k$-chromatic graph that is not $k$-choosable and shows a tight dichotomy: for odd $k$, every $k$-chromatic graph with $|V| \le 2k+2$ is chromatic-choosable, while for even $k$ only the two known graphs $K_{4, 2\star(k-1)}$ and $K_{3\star (k/2+1), 1\star (k/2-1)}$ can be non-$k$-choosable at $|V|=2k+2$. The authors extend the Ohba-type boundary through a sophisticated partition-and-matching framework, employing pseudo-$L$-colourings and near-acceptable colourings to derive a contradiction unless the graph is $k$-choosable. This yields a complete description of the minimal non-$k$-choosable complete multipartite graphs at size $2k+2$ and validates Noel’s conjecture in the odd-$k$ case. The methods combine Hall’s theorem, list-colouring techniques, and careful structural analysis of multipartite graphs to achieve a rigorous, constructive dichotomy with potential implications for related extremal list-colouring questions.

Abstract

A graph $G$ is called chromatic-choosable if $χ(G)=ch(G)$. A natural problem is to determine the minimum number of vertices in a $k$-chromatic non-$k$-choosable graph. It was conjectured by Ohba, and proved by Noel, Reed and Wu that $k$-chromatic graphs $G$ with $|V(G)| \le 2k+1$ are $k$-choosable. This upper bound on $|V(G)|$ is tight. It is known that if $k$ is even, then $G=K_{3 \star (k/2+1), 1 \star (k/2-1)}$ and $G=K_{4, 2 \star (k-1)}$ are $k$-chromatic graphs with $|V(G)| =2 k+2$ that are not $k$-choosable. Some subgraphs of these two graphs are also non-$k$-choosable. The main result of this paper is that all other $k$-chromatic graphs $G$ with $|V(G)| =2 k+2$ are $k$-choosable. In particular, if $χ(G)$ is odd and $|V(G)| \le 2χ(G)+2$, then $G$ is chromatic-choosable, which was conjectured by Noel.

Figures (1)

• Figure 1: The bipartite graph $B_f$ with partite sets $G_f$ and $C_L$. Vertices in $G_f$ are $f$-classes, some of them are singleton classes represented by solid circles, and other are $2^+$-classes, represented by solid squares. The broken arrowed line indicate the colouring $f$. The edges of $B_f$ are not drawn, and $Y_{\mathcal{S}} = N_{B_{\mathcal{S}}}(X_{\mathcal{S}})$. Vertex $v^*$ is contained in $X_{\mathcal{S}}$ but $f(v^*)=c^* \notin Y_{\mathcal{S}}$. So $v^{\star}$ is a badly $f$-coloured vertex.

Theorems & Definitions (37)

• Theorem 1.1: Noel-Reed-Wu Theorem
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Definition 1
• Lemma 2.1
• Lemma 3.2
• Theorem 4.1
• Claim 4.2
• Claim 4.3