On the Ohba Number and Generalized Ohba Numbers of Complete Bipartite Graphs

TL;DR

This work analyzes the Ohba number $\tau_0(a,b)$ and the generalized Ohba numbers $\tau_s(a,b)$ for complete bipartite graphs $K_{a,b}$, extending Noel's question N14. It proves $\tau_0(2,b)=\lfloor\sqrt{b}\rfloor-1$ for all $b\ge2$ and establishes $\tau_0(a,b)=\Omega(\sqrt{b})$ as $b\to\infty$ for fixed $a\ge2$, while developing lower and upper bounds for $\tau_s(a,b)$ in the generalized setting. The authors derive precise bounds when $a=s+2$ and provide exact values for select small $(s,b)$, illustrating the tightness of bounds and suggesting asymptotic behavior $\tau_s(a,b)=\Theta(b^{1/(2s+2)})$ under certain conditions. The results deepen understanding of chromatic-choosability via graph joins and raise open questions about tightness and asymptotics for broader parameter ranges.

Abstract

We say that a graph $G$ is chromatic-choosable when its list chromatic number $χ_{\ell}(G)$ is equal to its chromatic number $χ(G)$. Chromatic-choosability is a well-studied topic, and in fact, some of the most famous results and conjectures related to list coloring involve chromatic-choosability. In 2002 Ohba showed that for any graph $G$ there is an $N \in \mathbb{N}$ such that the join of $G$ and a complete graph on at least $N$ vertices is chromatic-choosable. The Ohba number of $G$ is the smallest such $N$. In 2014, Noel suggested studying the Ohba number, $τ_{0}(a,b)$, of complete bipartite graphs with partite sets of size $a$ and $b$. In this paper we improve a 2009 result of Allagan by showing that $τ_{0}(2,b) = \lfloor \sqrt{b} \rfloor - 1$ for all $b \geq 2$, and we show that for $a \geq 2$, $τ_{0}(a,b) = Ω( \sqrt{b} )$ as $b \rightarrow \infty$. We also initiate the study of some relaxed versions of the Ohba number of a graph which we call generalized Ohba numbers. We present some upper and lower bounds of generalized Ohba numbers of complete bipartite graphs while also posing some questions.

Theorems & Definitions (46)

• Theorem 1
• Theorem 2: NR15
• Lemma 3
• proof
• Theorem 4: A09
• Lemma 5
• Theorem 6
• Corollary 7
• Theorem 8
• Theorem 9