有向色数中的MAD相变
MAD Phase Transitions in the Oriented Chromatic Number
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中文总结 AI 辅助
研究有向图的有向色数，通过证明对于足够大阶数的\(d -\)退化图，给出其有向色数的上界，得出对于固定\(r\geq 4\)，最大平均度小于\(r\)的图的有向色数最优界为\(\Theta(\sqrt{n})\)，补充了相关研究。
中文摘要 AI 辅助
对于有向图\(G\)，其有向色数\(\chi_o(G)\)是使得\(G\)同态于\(t\)个顶点的竞赛图的最小整数\(t\)。简单图\(H\)的有向色数是\(H\)所有定向中的最大有向色数。1999年Borodin等人证明对于所有\(\epsilon>0\)，最大平均度小于\(4 - \epsilon\)的图有界有向色数。本文证明对于每个正整数\(d\)，每个足够大阶数的\(d -\)退化图\(G\)满足\(\chi_o(G) \leq 6(1+\frac{9d^2}{4})^{\frac{1}{2}}d 8^{d}\sqrt{n}\)。这意味着对于固定\(r\geq 4\)，最大平均度小于\(r\)的图的有向色数的最优界是\(\Theta(\sqrt{n})\)，补充了Wood的界。
英文摘要
For an oriented graph $G$ the oriented chromatic number of $G$, written $χ_o(G)$, is the least integer $t$ such that $G$ has a homomorphism to a tournament on $t$ vertices. The oriented chromatic number of a simple graph $H$ is the maximum oriented chromatic number over all orientations of $H$. Borodin, Kostochka, Ne{š}et{ř}il, Raspaud, and Sopena proved in 1999 that for all $ε>0$, graphs with maximum average degree less than $4-ε$ have bounded oriented chromatic number. This is in some sense optimal, because $1$-subdivisions of cliques demonstrate that there exists graphs with maximum average degree strictly less than $4$ and oriented chromatic number $Ω(\sqrt{n})$. We prove that for every positive integer $d$, every $d$-degenerate graph $G$ with sufficiently large order satisfies $χ_o(G) \leq 6(1+\frac{9d^2}{4})^{\frac{1}{2}}d 8^{d}\sqrt{n}$. This implies for a fixed $r\geq 4$, the optimal bound for the oriented chromatic number of graphs with maximum average degree less than $r$ is $Θ(\sqrt{n})$. This complements a bound of Wood, who showed that for all $n$ vertex graphs $χ_o \leq 2Δ\sqrt{n-1}$ where $Δ$ is the maximum degree.
