Intersections of Cycles in $k$-Connected Graphs

Abstract

Let $k\ge 2$ be a positive integer and $G$ be a $k$-connected graph. We prove that for any two cycles $C$ and $D$ in $G$ there exist two cycles $C^*$ and $D^*$ such that $V(C)\cup V(D)\subseteq V(C^*)\cup V(D^*)$ and $|V(C^*)\cap V(D^*)| \ge \frac{1}{\left( \root 3 \of {256} +3\right) ^{3/5}} \cdot k^{3/5}$. Moreover, we conjecture that $\frac{1}{\left( \root 3 \of {256} +3\right) ^{3/5}} \cdot k^{3/5}$ can be replaced by $k$ and verify this conjecture for $k \le 6$. For $\alpha > k$, Fouquet and Jolivet conjectured that if a $k$-connected graph $G$ with an independent number $\alpha$ has a cycle of length at least $k(n+\alpha -k)/\alpha$. Based on the proof techniques used by Manoussakis and by Chen, Hu and Wu, we believed that resolving our conjecture will provide some help on tackling the Fouquet–Jolivet conjecture.