A Note on Fractional DP-Coloring of Graphs
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2019
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| _version_ | 1866910458248691712 |
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| author | Dominik, Daniel Kaul, Hemanshu Mudrock, Jeffrey A. |
| author_facet | Dominik, Daniel Kaul, Hemanshu Mudrock, Jeffrey A. |
| contents | DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvořák and Postle in 2015. In 2019, Bernshteyn, Kostochka, and Zhu introduced a fractional version of DP-coloring. They showed that unlike the fractional list chromatic number, the fractional DP-chromatic number of a graph $G$, denoted $χ_{_{DP}}^*(G)$, can be arbitrarily larger than $χ^*(G)$, the graph's fractional chromatic number. We generalize a result of Alon, Tuza, and Voigt (1997) on the fractional list chromatic number of odd cycles, and, in the process, show that for each $k \in \mathbb{N}$, $χ_{_{DP}}^*(C_{2k+1}) = χ^*(C_{2k+1})$. We also show that for any $n \geq 2$ and $m \in \mathbb{N}$, if $p^*$ is the solution in $(0,1)$ to $p=(1-p)^n$ then $χ_{_{DP}}^*(K_{n,m})\leq1/p^*$, and we prove a generalization of this result for multipartite graphs. Finally, we determine a lower bound on $χ_{_{DP}}^*(K_{2,m})$ for any $m \geq 3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_03416 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | A Note on Fractional DP-Coloring of Graphs Dominik, Daniel Kaul, Hemanshu Mudrock, Jeffrey A. Combinatorics 05C15, 05C69 DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvořák and Postle in 2015. In 2019, Bernshteyn, Kostochka, and Zhu introduced a fractional version of DP-coloring. They showed that unlike the fractional list chromatic number, the fractional DP-chromatic number of a graph $G$, denoted $χ_{_{DP}}^*(G)$, can be arbitrarily larger than $χ^*(G)$, the graph's fractional chromatic number. We generalize a result of Alon, Tuza, and Voigt (1997) on the fractional list chromatic number of odd cycles, and, in the process, show that for each $k \in \mathbb{N}$, $χ_{_{DP}}^*(C_{2k+1}) = χ^*(C_{2k+1})$. We also show that for any $n \geq 2$ and $m \in \mathbb{N}$, if $p^*$ is the solution in $(0,1)$ to $p=(1-p)^n$ then $χ_{_{DP}}^*(K_{n,m})\leq1/p^*$, and we prove a generalization of this result for multipartite graphs. Finally, we determine a lower bound on $χ_{_{DP}}^*(K_{2,m})$ for any $m \geq 3$. |
| title | A Note on Fractional DP-Coloring of Graphs |
| topic | Combinatorics 05C15, 05C69 |
| url | https://arxiv.org/abs/1910.03416 |