Maximum Number of Contractible Edges on Longest Cycles of a $3$-Connected Graph

Abstract

We show that if $G$ is a $3$-connected graph of order at least $5$, then there exists a longest cycle $C$ of $G$ such that the number of contractible edges of $G$ which are on $C$ is greater than or equal to $\frac{|V(C)|+9}{8}.$