Monochromatic cycle partitions of $2$-coloured graphs with minimum degree $3n/4$

TL;DR

This paper proves a conjecture that large graphs with minimum degree 3n/4 can be partitioned into two monochromatic cycles, one of each colour, for sufficiently large n.

Contribution

It establishes the conjecture for large graphs, improving previous approximate results and confirming the partitioning property.

Findings

Confirmed the conjecture for sufficiently large n

Demonstrated that such graphs can be partitioned into two monochromatic cycles

Improved upon previous approximate results

Abstract

Balogh, Bar\'at, Gerbner, Gy\'arf\'as, and S\'ark\"ozy proposed the following conjecture. Let $G$ be a graph on $n$ vertices with minimum degree at least $3n/4$. Then for every $2$-edge-colouring of $G$, the vertex set $V(G)$ may be partitioned into two vertex-disjoint cycles, one of each colour. We prove that this conjecture holds for $n$ large enough, improving approximate results by the aforementioned authors and by DeBiasio and Nelsen.