Monochromatic cycles and the monochromatic circumference in 2-coloured graphs

TL;DR

This paper proves a conjecture about the existence of monochromatic cycles of various lengths in large 2-edge coloured graphs with high minimum degree, and characterizes extremal graphs lacking such cycles.

Contribution

It confirms the conjecture for large graphs and identifies all extremal graphs with minimum degree exactly 3n/4 that lack certain monochromatic cycles.

Findings

Graphs with minimum degree > 3n/4 contain all monochromatic cycles of length 4 to n/2.

Characterization of extremal graphs with minimum degree = 3n/4 lacking certain cycles.

High minimum degree guarantees long monochromatic cycles or cycles of all lengths in one colour.

Abstract

Li, Nikiforov and Schelp conjectured that a 2-edge coloured graph G with order n and minimal degree strictly greater than 3n/4 contains a monochromatic cycle of length l, for all l at least four and at most n/2. We prove this conjecture for sufficiently large n and also find all 2-edge coloured graphs with minimal degree equal to 3n/4 that do not contain all such cycles. Finally we show that, for all positive constants d and sufficiently large n, a 2-edge coloured graph G of order n with minimal degree at least 3n/4 either contains a monochromatic cycle of length at least (2/3+d/2)n, or, in one of the two colours, contains a cycle of all lengths between three and (2/3-d)n.