Long monochromatic even cycles in 3-edge-coloured graphs of large minimum degree

TL;DR

This paper proves that in large 3-edge-coloured graphs with high minimum degree, there always exists a monochromatic even cycle of a specified length, extending understanding of monochromatic structures in dense graphs.

Contribution

It establishes a new threshold for minimum degree ensuring monochromatic even cycles in 3-coloured graphs, advancing extremal graph theory results.

Findings

Monochromatic even cycles exist under specified degree conditions

Thresholds depend on the number of vertices and minimum degree

Results apply to large graphs with three-edge colourings

Abstract

We show that for every $\eta>0$, there exists $n_0$ such that for every even $n$, $n\ge n_0$, and every graph $G$ with $(2+\eta)n$ vertices and minimum degree at least $(7/4+4\eta)n$, each colouring of the edges of $G$ with three colours results in a monochromatic cycle of length $n$.