最小化圈长的调和和

中文摘要 AI 辅助

极值图论的一个核心主题是理解图的密度与其圈长谱（即图中出现的不同圈长的集合）丰富度之间的关系。1966年，Erdős和Hajnal建议研究$s(G):=\sum_{\ell\in{C}(G)}1/\ell$作为图$G$的圈长谱${C}(G)$丰富度的度量。通过一系列越来越强的猜想，Erdős提出在具有相同平均度的所有图$G$中，完全二分图使$s(G)$最小。其中最尖锐的猜想来自1981年，它指出在具有至少$k(n-k)$条边的所有$n$顶点图中（其中$k\leq n/2$），图$K_{k, n-k}$使$s(G)$最小。我们通过证明更强的陈述来证明该猜想对所有足够大的$k$成立：任何满足$e(G)>(k-1)(n-k+1)$且$n\geq 2k$的$n$顶点图$G$都满足$s(G)\geq\sum_{\ell=2}^{k}1/(2\ell)$。此外，我们证明完全二分图$K_{k,n-k}$是具有至少$k(n-k)$条边且在此处达到等号的唯一图。

英文摘要

A central theme in extremal graph theory is to understand the relationship between the density of a graph and the richness of its cycle length spectrum, which is the set of distinct cycle lengths occurring in the graph. In 1966, Erdős and Hajnal suggested studying $s(G):=\sum_{\ell\in{C}(G)}1/\ell$ as a measure of the richness of the cycle length spectrum ${C}(G)$ of a graph $G$. Through a series of increasingly strong conjectures, Erdős suggested that the complete bipartite graphs minimise $s(G)$ among all graphs $G$ with the same average degree. The sharpest such conjecture, from 1981, states that the graph $K_{k, n-k}$ minimises $s(G)$ among all $n$-vertex graphs with at least $k(n-k)$ edges (where $k\leq n/2$). We prove this conjecture for all sufficiently large $k$, by showing the stronger statement that any $n$-vertex graph $G$ with $e(G)>(k-1)(n-k+1)$ and $n\geq 2k$ satisfies $s(G)\geq\sum_{\ell=2}^{k}1/(2\ell)$. Moreover, we show that the complete bipartite graph $K_{k,n-k}$ is the unique graph with at least $k(n-k)$ edges that achieves equality here.

发表机构

• ETH, Zürich（苏黎世联邦理工学院）
• University of Warwick（华威大学）
• University College London（伦敦大学学院）

机构由 AI 辅助整理，请以论文原文为准。