Intersection of cycles and paths in k-connected graphs

Abstract

McGuinness (2005) shows that if $G$ is a $k$-connected graph $(k\ge 2)$ having circumference $c=c\left(G\right)\ge 2k$, then for a pair of cycles $C$ and $D$ of $G$ such that $\left|V\left(C\right)\right|+\left|V\left(D\right)\right|\ge 2c-2k+3$, it must be true that $C$ and $D$ intersect in at least two common vertices. Using this result, McGuinness proves that for any $k$-connected graph $G$ where $k\ge 2$ and having circumference $c\ge 2k$, there is a bond $B$ which intersects every cycle of length $c-k+2$ or greater.

In this paper, we study the following general questions: will two long cycles or two paths intersect at a large number of vertices in a highly connected graph? We give positive answers to both questions and extend McGuinness’ result. Let $c\left(G\right)$ denote the circumference of a graph $G$. We prove the following results.

(1) Let $G$ be a $k$-connected graph for $s\ge 3$ and $k\ge 13.9413s^2$. Suppose $C$ and $D$ are cycles of $G$ such that $\left|V\left(C\right)\right|+\left|V\left(D\right)\right|\ge 2c\left(G\right)-\frac{c\sqrt{k}}{s}+12$ where $c=2\sqrt{\frac{13}{7}}\approx 2.7255$. Then $C$ and $D$ meet in at least $s+1$ common vertices.

(2) Let $G$ be a $k$-connected graph where $k\ge 2710$. Suppose $C$ and $D$ are cycles such that $\left|V\left(C\right)\right|+\left|V\left(D\right)\right|\ge 2c\left(G\right)-c\sqrt[6]{k}+12$ where $c=2\sqrt{\frac{13}{7}}\approx 2.7255$. Then $C$ and $D$ meet in at least $\lfloor \sqrt[3]{k}\rfloor +1$ common vertices.

We also prove similar results on the intersection of long paths for $k$-connected graphs.