给定奇围长与大代数连通度的图的结构

中文摘要 AI 辅助

Andrásfai、Erdős和Sós的经典结论指出，每个n顶点、奇围长至少为2k+1且最小度大于2n/(2k+1)的图都是二分图。本文不施加最小度条件，转而研究代数连通度对给定奇围长的图的简单结构的约束条件。图G的代数连通度记为μ₂(G)，是其拉普拉斯矩阵的第二小特征值。主要结果如下：1. 每个n顶点、无三角形且μ₂(G)≥n/3的图G都是二分图，且常数1/3是渐近最优的；2. 对于k≥3，每个n顶点、奇围长至少为2k+1且μ₂(G)>4n/(6k-1)的图G都是二分图；3. 对于k≥22，每个n顶点、奇围长至少为2k+1且μ₂(G)>3456n/k³的图G都是二分图，且项k⁻³是渐近最优的。

英文摘要

A classical result of Andrásfai, Erdős, and Sós states that every $n$-vertex graph with odd girth at least $2k+1$ and minimum degree larger than $\frac{2n}{2k+1}$ is bipartite. Rather than imposing a minimum-degree condition, in this paper we investigate conditions on algebraic connectivity that force graphs of given odd girth to have a simple structure. The algebraic connectivity of a graph $G$, denoted by $μ_2(G)$, is the second smallest eigenvalue of its Laplacian matrix. Our main results are as follows. 1. Every $n$-vertex triangle-free graph $G$ with $μ_2(G)\geq \frac{n}{3}$ is bipartite. Moreover, the constant $\frac{1}{3}$ is asymptotically best possible. 2. For $k\geq 3$, every $n$-vertex graph $G$ of odd girth at least $2k+1$ with $μ_2(G)>\frac{4n}{6k-1}$ is bipartite. 3. For $k\geq 22$, every $n$-vertex graph $G$ of odd girth at least $2k+1$ with $μ_2(G)>\frac{3456n}{k^3}$ is bipartite. Moreover, the term $k^{-3}$ is asymptotically best possible.

发表机构

• Shanghai Jiao Tong University（上海交通大学）
• Nankai University（南开大学）

机构由 AI 辅助整理，请以论文原文为准。