On the Ohba Number and Generalized Ohba Numbers of Complete Bipartite Graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866915811375972352 |
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| author | Cano, Kennedy Gutknecht, Emily Kappaganthula, Gautham Miller, George Mudrock, Jeffrey A. Thornburgh, Ezekiel |
| author_facet | Cano, Kennedy Gutknecht, Emily Kappaganthula, Gautham Miller, George Mudrock, Jeffrey A. Thornburgh, Ezekiel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_06291 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Ohba Number and Generalized Ohba Numbers of Complete Bipartite Graphs Cano, Kennedy Gutknecht, Emily Kappaganthula, Gautham Miller, George Mudrock, Jeffrey A. Thornburgh, Ezekiel Combinatorics 05C15 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. |
| title | On the Ohba Number and Generalized Ohba Numbers of Complete Bipartite Graphs |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2403.06291 |