Graphs with Almost All Edges in Long Cycles

Abstract

For an edge e of a given graph G, define $c_e(G)$ be the length of a longest cycle of G containing e. Wang and Lv (2008) gave a tight function $f_{0}(n,k)$ (for integers $n\ge 3$ and $k\ge 4$), such that for any 2-connected graph G on n vertices with more than $f_{0}(n,k)$ edges, every edge belongs to a cycle of length at least k, i.e., $c_e(G)\ge k$ for every edge $e\in E(G)$. In this work we give a tight function f(n, k) (for integers $n\ge k\ge 6$), such that for any 2-connected graph G on n vertices with more than f(n, k) edges, we have that $c_e(G)\ge k$ for all but at most one edge of G.