A Note on Circumferences in $k$-Connected Graphs with Given Independence Number

Abstract

Fouquet and Jolivet conjectured that if $G$ is a $k$-connected $n$-vertex graph with independence number $\alpha \ge k\ge 2$, then $G$ has circumference at least $\frac{k(n+\alpha -k)}{\alpha }$. This conjecture was recently proved by $O$, West, and Wu.
In this note, we consider the set of $k$-connected $n$-vertex graphs with independence number $\alpha >k\ge 2$ and circumference exactly $\frac{k(n+\alpha -k)}{\alpha }$. We show that all of these graphs have a similar structure.