An edge-coloured version of Dirac's theorem

TL;DR

This paper proves that edge-coloured graphs with sufficiently high minimum colour degree contain properly coloured 2-factors and cycles of all lengths up to the number of vertices, extending Dirac's theorem to edge-coloured graphs.

Contribution

It establishes a threshold for the minimum colour degree ensuring the existence of properly coloured 2-factors and cycles of all lengths, generalizing Dirac's theorem to edge-coloured graphs.

Findings

Graphs with δ^c(G) ≥ 2|G|/3 contain properly coloured 2-factors.

Graphs with δ^c(G) ≥ (2/3 + ε)|G| contain properly coloured cycles of all lengths.

The bounds are tight; below 2n/3, the property does not hold.

Abstract

Let $G$ be an edge-coloured graph. The minimum colour degree $ \delta^c(G) $ of $G$ is the largest integer $k$ such that, for every vertex $v$, there are at least $k$ distinct colours on edges incident to $v$. We say that $G$ is properly coloured if no two adjacent edges have the same colour. In this paper, we show that every edge-coloured graph $G$ with $ \delta^c(G) \ge 2|G| / 3 $ contains a properly coloured $2$-factor. Furthermore, we show that for any $ \varepsilon > 0 $ there exists an integer $ n_0 $ such that every edge-coloured graph $G$ with $|G| = n \ge n_0 $ and $ \delta^c(G) \ge ( 2/3 + \varepsilon ) n $ contains a properly coloured cycle of length $\ell$ for every $3 \le \ell \le n$. This result is best possible in the sense that the statement is false for $ \delta^c(G) < 2n / 3 $.