On a Conjecture of Fan Concerning Average Degrees and Long Cycles

Abstract

In this paper, we prove the following result:

Let $k$ be an integer with $k\ge 5$. Let $C$ be a cycle in a graph $G$, and let $H$ be a component of $G-C$. Suppose that $C$ is locally longest with respect to $H$, and $H$ is locally $k$-connected to $C$, $\left|V(H)\right|\ge k-1$, $\delta (H)\ge \lfloor (k-1)/2\rfloor$, and $H$ is 3-connected. Let $r=({\sum }_{x\in V(H)}{\mathrm{deg}}_G(x))/|V(H)|$. Then $l(C)\ge k(r+2-k)$, with equality only if $r$ is an integer and either $H$ is a complete graph of order $r+1-k$ and every vertex of $H$ has the same $k$ neighbours on $C$, or $H$ is a complete graph of order $k-1$ and every vertex of $H$ has the same $r+2-k$ neighbours on $C$.