On two conjectures about the intersection of longest paths and cycles

TL;DR

This paper investigates conjectures about the intersection sizes of longest paths and cycles in k-connected graphs, providing new bounds and confirming the conjecture for certain connectivity and size conditions.

Contribution

It establishes a new lower bound for the intersection of longest cycles in k-connected graphs and relates conjectures for paths and cycles, confirming their validity under specific parameters.

Findings

Confirmed Smith's conjecture for k ≥ (n+16)/7.

Established a lower bound of min{n, 8k - n - 16} for cycle intersections.

Related path and cycle intersection conjectures, validating Hippchen's conjecture for certain k values.

Abstract

A conjecture attributed to Smith states that every pair of longest cycles in a $k$-connected graph intersect each other in at least $k$ vertices. In this paper, we show that every pair of longest cycles in a~$k$-connected graph on $n$ vertices intersect each other in at least~$\min\{n,8k-n-16\}$ vertices, which confirms Smith's conjecture when $k\geq (n+16)/7$. An analog conjecture for paths instead of cycles was stated by Hippchen. By a simple reduction, we relate both conjectures, showing that Hippchen's conjecture is valid when either $k \leq 6$ or $k \geq (n+9)/7$.