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- W3205614114 abstract "The proper chromatic number $Vec{chi}(G)$ of a graph $G$ is the minimum $k$ such that there exists an orientation of the edges of $G$ with all vertex-outdegrees at most $k$ and such that for any adjacent vertices, the outdegrees are different. Two major conjectures about the proper chromatic number are resolved. First it is shown, that $Vec{chi}(G)$ of any planar graph $G$ is bounded (in fact, it is at most 14). Secondly, it is shown that for every graph, $Vec{chi}(G)$ is at most $O(frac{rlog r}{loglog r})+tfrac{1}{2}MAD(G)$, where $r=chi(G)$ is the usual chromatic number of the graph, and $MAD(G)$ is the maximum average degree taken over all subgraphs of $G$. Several other related results are derived. Our proofs are based on a novel notion of fractional orientations." @default.
- W3205614114 created "2021-10-25" @default.
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- W3205614114 date "2021-10-13" @default.
- W3205614114 modified "2023-10-17" @default.
- W3205614114 title "Proper orientations and proper chromatic number" @default.
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- W3205614114 doi "https://doi.org/10.48550/arxiv.2110.07005" @default.
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