On the Circumference of 3-Connected Cubic Triangle-Free Plane Graphs

Abstract

The circumference of a graph $G$ is the length of a longest cycle in $G$ , denoted by $\text{cir}\left(G\right)$ . For any even number $n$ , let $c\left(n\right)$  = min { $\text{cir}\left(G\right)|G$ is a 3-connected cubic triangle-free plane graph with $n$ vertices}. In this paper, we show that an upper bound of $c\left(n\right)$ is $n+1-3\lfloor n/136\rfloor$ for $n\ge 136$ .