Fractional Strict Degeneracy of Graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917453319110656 |
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| author | Dominik, Daniel Mudrock, Jeffrey A. |
| author_facet | Dominik, Daniel Mudrock, Jeffrey A. |
| contents | DP-coloring (also called correspondence coloring) is a generalization of list coloring introduced by Dvořák and Postle in 2015. The DP-chromatic number of a graph $G$, $χ_{_{DP}}(G)$, is the analogue of the chromatic number of $G$ in the DP context and is bounded above by the degeneracy of $G$ plus one. Over the last two years a plethora of authors have introduced variations on the notion of degeneracy and used these new ideas to give improved bounds on the DP-chromatic number of certain families of graphs.
Fractional DP-coloring is a generalization of fractional list coloring introduced by Bernshteyn, Kostochka, and Zhu in 2019. In this paper we introduce two analogues of the degeneracy of a graph to the fractional context, each of which bound its fractional DP-chromatic number from above. We use these analogues to bound the fractional DP-chromatic number of a variety of graphs including unicyclic graphs, some complete bipartite graphs, and sparse graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13212 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fractional Strict Degeneracy of Graphs Dominik, Daniel 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. The DP-chromatic number of a graph $G$, $χ_{_{DP}}(G)$, is the analogue of the chromatic number of $G$ in the DP context and is bounded above by the degeneracy of $G$ plus one. Over the last two years a plethora of authors have introduced variations on the notion of degeneracy and used these new ideas to give improved bounds on the DP-chromatic number of certain families of graphs. Fractional DP-coloring is a generalization of fractional list coloring introduced by Bernshteyn, Kostochka, and Zhu in 2019. In this paper we introduce two analogues of the degeneracy of a graph to the fractional context, each of which bound its fractional DP-chromatic number from above. We use these analogues to bound the fractional DP-chromatic number of a variety of graphs including unicyclic graphs, some complete bipartite graphs, and sparse graphs. |
| title | Fractional Strict Degeneracy of Graphs |
| topic | Combinatorics 05C15, 05C69 |
| url | https://arxiv.org/abs/2604.13212 |