Ramsey Numbers of Trails

TL;DR

This paper studies the Ramsey numbers of trails in graphs, establishing bounds that improve upon trivial estimates and providing new insights into the minimal graph sizes needed to guarantee trail structures.

Contribution

It introduces the concept of Ramsey numbers of trails and proves new upper and lower bounds, advancing understanding in graph Ramsey theory.

Findings

Ramsey number of trails with k vertices is at most k.

Lower bound of the Ramsey number is 2√k + Θ(1).

Improves upon the trivial upper bound of 3k/2 - 1.

Abstract

We initiate the study of Ramsey numbers of trails. Let $k \geq 2$ be a positive integer. The Ramsey number of trails with $k$ vertices is defined as the the smallest number $n$ such that for every graph $H$ with $n$ vertices, $H$ or the complete $\overline{H}$ contains a trail with $k$ vertices. We prove that the Ramsey number of trails with $k$ vertices is at most $k$ and at least $2\sqrt{k}+\Theta(1)$. This improves the trivial upper bound of $\lfloor 3k/2\rfloor -1$.