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arXiv 2107.09290math.CO
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边标号完全图中的不平衡生成子图

Unbalanced spanning subgraphs in edge labeled complete graphs

Stéphane Bessy, Johannes Pardey, Lucas Picasarri-Arrieta, Dieter Rautenbach

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摘要：

设$K$是阶为$n$的完全图。对于$d\in (0,1)$，令$c$是$K$的$\pm 1$ -边标记，使得存在具有标记$+1$的$d{n\choose 2}$边，并且令$G$是$K$的最大度至多为$Δ$的生成子图。我们证明了存在一个同构的副本$G'$的$G$在$K$，使边缘的数量与标签$+1$在$G'$至少是$\left(c_{d,Δ}-O\left(\frac{1}{n}\right)\right)m(G)$，其中$c_{d,Δ}=d+Ω\left(\frac{1}Δ\right)$固定的$d$，也就是说，这个数字明显偏离其预期值时，考虑一个均匀随机副本的$G$在$K$。对于$d=\frac{1}{2}$和$Δ\leq 2$，我们提供了更详细的结果。

英文摘要：

Let $K$ be a complete graph of order $n$. For $d\in (0,1)$, let $c$ be a $\pm 1$-edge labeling of $K$ such that there are $d{n\choose 2}$ edges with label $+1$, and let $G$ be a spanning subgraph of $K$ of maximum degree at most $Δ$. We prove the existence of an isomorphic copy $G'$ of $G$ in $K$ such that the number of edges with label $+1$ in $G'$ is at least $\left(c_{d,Δ}-O\left(\frac{1}{n}\right)\right)m(G)$, where $c_{d,Δ}=d+Ω\left(\frac{1}Δ\right)$ for fixed $d$, that is, this number visibly deviates from its expected value when considering a uniformly random copy of $G$ in $K$. For $d=\frac{1}{2}$, and $Δ\leq 2$, we present more detailed results.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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