Abstract
Balogh, Barát, Gerbner, Gyárfás, and Sárközy made 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 this conjecture for large $n$, improving approximate results by the aforementioned authors and by DeBiasio and Nelsen.
Highlights
While undergraduates in Budapest, Gerencser and Gyarfas [11] proved the following simple result: for any 2-edge-colouring of the complete graph Kn, there exists a monochromatic path of length at least 2n/3
Gerencser and Gyarfas observe that a weaker result, asserting the existence of a monochromatic path of length at least n/2, can be deduced from the following simple observation: for any red and blue colouring of Kn, there is a Hamilton path which is the union of a red path and a blue path
Lehel went even further: he conjectured that for every 2-colouring of Kn the vertex set may be partitioned into two monochromatic cycles of distinct colours
Summary
While undergraduates in Budapest, Gerencser and Gyarfas [11] proved the following simple result: for any 2-edge-colouring of the complete graph Kn, there exists a monochromatic path of length at least 2n/3. Gerencser and Gyarfas observe that a weaker result, asserting the existence of a monochromatic path of length at least n/2, can be deduced from the following simple observation: for any red and blue colouring of Kn, there is a Hamilton path which is the union of a red path and a blue path. We remark that in our context, the empty set, a single vertex and an the electronic journal of combinatorics 26(1) (2019), #P1.19 edge are considered to be cycles This conjecture first appeared in [2], where it was proved for some special colourings of Kn. In 1998, almost twenty years after this conjecture was made, Luczak, Rodl and Szemeredi [22] proved it for large n, using the regularity lemma. Lehel’s conjecture was fully resolved in 2010 by Bessy and Thomasse [5] with an elegant and short proof
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
