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arXiv · 2403.06291

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

Kennedy Cano·Emily Gutknecht·Gautham Kappaganthula·George Miller·Jeffrey A. Mudrock·Ezekiel Thornburgh
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Abstract

We say that a graph $G$ is chromatic-choosable when its list chromatic number $\chi_{\ell}(G)$ is equal to its chromatic number $\chi(G)$. 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, $\tau_{0}(a,b)$, 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 $\tau_{0}(2,b) = \lfloor \sqrt{b} \rfloor - 1$ for all $b \geq 2$, and we show that for $a \geq 2$, $\tau_{0}(a,b) = \Omega( \sqrt{b} )$ 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.

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Kennedy Cano, Emily Gutknecht, Gautham Kappaganthula, George Miller, Jeffrey A. Mudrock, Ezekiel Thornburgh. 2024-03-10. On the Ohba Number and Generalized Ohba Numbers of Complete Bipartite Graphs. https://arxiv.org/abs/2403.06291

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