带权循环局部化不等式
A weighted cycle-localization inequality
AI总结：
本文研究带权图的循环局部化不等式，提出加权版本的Bondy-Fan定理，并探讨等号成立的图类及必要条件。
摘要：
1959年，Erdős和Gallai证明了每2-连通图包含长度至少为2|E(G)|/|V(G)|-1的环。Bondy和Fan于1991年将其扩展到加权图。本文建立了一个加权局部版本的不等式：对于加权图(G,w)，设C_w(e)为包含e的最长环的权重（桥设为2w(e)），则∑_{e∈E(G)}w(e)/C_w(e) ≤ (n-1)/2。该结果是Bondy-Fan定理的加权局部类比，建立了全局与局部视角的对应关系，并给出了等号成立的图类及必要条件。
英文摘要：
In 1959, Erdős and Gallai showed that every $2$-connected graph $G$ contains a cycle of length at least $\frac{2|E(G)|}{|V(G)|-1}$. This result was subsequently extended to weighted graphs by Bondy and Fan in 1991. A natural local variant of this problem arises by considering, for each edge $e\in E(G)$, the quantity $c(e)$, defined as the length of the longest cycle in $G$ containing $e$ (with $c(e)=2$ if $e$ is a bridge). Zhao and Zhang recently proved that for every graph $G$ on $n$ vertices satisfies $\sum_{e\in E(G)}\frac{1}{c(e)}\le \frac{n-1}{2}.$ In this note, we establish a weighted generalization of this inequality. For a weighted graph $(G,w)$ with positive edge weights, let $C_w(e)$ denote the maximum weight of a cycle containing $e$ (setting $C_w(e)=2w(e)$ if $e$ is a bridge). We prove that $$ \sum_{e\in E(G)}\frac{w(e)}{C_w(e)}\le \frac{n-1}{2}. $$ Our result can be viewed as a weighted local analogue of the Bondy-Fan theorem, thereby establishing a correspondence between the global and local perspectives. Furthermore, we present a broad class of graphs attaining equality and derive necessary conditions for equality.
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