# Monochromatic paths and cycles in $2$-edge-colored graphs with large minimum degree

## TL;DR

This paper proves that large graphs with high minimum degree contain monochromatic cycles of many lengths under any 2-edge-coloring, confirming several conjectures and extending classical results in graph Ramsey theory.

## Contribution

It establishes tight minimum degree conditions ensuring monochromatic cycles of all lengths or all even lengths, confirming conjectures by Schelp and others for large graphs.

## Key findings

- Graphs with minimum degree ≥ (3n-1)/4 contain monochromatic cycles of all lengths up to 2t+r.
- The results confirm Schelp's conjecture for large graphs regarding monochromatic cycles.
- The findings imply that such graphs also contain long monochromatic paths and cycles of length at least 2t+r.

## Abstract

A graph $G$ arrows a graph $H$ if in every $2$-edge-coloring of $G$ there exists a monochromatic copy of $H$. Schelp had the idea that if the complete graph $K_n$ arrows a small graph $H$, then every "dense" subgraph of $K_n$ also arrows $H$, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large $n$, if $n = 3t+r$ where $r \in \{0,1,2\}$ and $G$ is an $n$-vertex graph with $\delta(G) \ge (3n-1)/4$, then for every $2$-edge-coloring of $G$, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same color, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the same color. Our result is tight in the sense that no longer cycles (of length $>2t+r$) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large $n$, every $(3t-1)$-vertex graph $G$ with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with $2n$ vertices. Moreover, it implies for sufficiently large $n$ the conjecture by Benevides, {\L}uczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every $n$-vertex graph $G$ with $\delta(G) \ge 3n/4$, in each $2$-edge-coloring of $G$ there exists a monochromatic cycle of length at least $2t+r$.