有界缺陷正常定向与曲面上的图

中文摘要 AI 辅助

Chen、Mohar 和 Wu [J. Combin. Theory Ser. B 161 (2023)] 猜想每个可定向亏格为 $g$ 的图都有正常定向数 $O(\sqrt g)$。我们证明了这个猜想。我们的主要结果将潜在出度方法从独立颜色类扩展到三个有界内部度的部分。设 $k,D\in\mathbb Z_{\ge0}$。如果 $G$ 的最大平均度至多为 $2k$ 且 $V(G)=V_1\cup V_2\cup V_3$，其中每个 $V_i$ 诱导一个最大度至多为 $D$ 的子图，则 $$\pchi(G)\le k+9D+6+\floor{\frac32\ceil{\frac{2D+2}{5}}}\le k+\floor{\frac{48D}{5}}+8.$$ 当划分给定时，我们能在多项式时间内构造这样的定向。对于每个 $D\ge1$，我们还构造了一个图，其划分为三个内部最大度至多 $D$ 的部分，且 $\pchi(G)-\ceil{\mad(G)/2}\ge6D+5$；因此对 $D$ 的线性依赖是不可避免的。最后，Woodall 的缺陷着色定理得出：对于每个欧拉亏格为 $\gamma$ 的图，$\pchi(G)=O(\sqrt{\gamma+1})$。

英文摘要

Chen, Mohar and Wu [J. Combin. Theory Ser. B 161 (2023)] conjectured that every graph of orientable genus $g$ has proper orientation number $O(\sqrt g)$. We prove this conjecture. Our main result extends the potential-outdegree method from independent color classes to three parts of bounded internal degree. Let $k,D\in\mathbb Z_{\ge0}$. If $G$ has maximum average degree at most $2k$ and $V(G)=V_1\cup V_2\cup V_3$, where each $V_i$ induces a subgraph of maximum degree at most $D$, then $$\pchi(G)\le k+9D+6+\floor{\frac32\ceil{\frac{2D+2}{5}}}\le k+\floor{\frac{48D}{5}}+8.$$ When the partition is given, we construct such an orientation in polynomial time. For every $D\ge1$, we also construct a graph with a partition into three parts of internal maximum degree at most $D$ for which $\pchi(G)-\ceil{\mad(G)/2}\ge6D+5$; hence the linear dependence on $D$ is unavoidable. Finally, Woodall's defective-coloring theorem yields $\pchi(G)=O(\sqrt{γ+1})$ for every graph of Euler genus $γ$.

发表机构

• School of Mathematics and Statistics, Jiangsu Normal University（江苏师范大学数学与统计学院）

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