A Dirac type condition for properly coloured paths and cycles

TL;DR

This paper establishes a Dirac-type condition for edge-coloured graphs, guaranteeing the existence of long properly coloured paths or cycles based on local colour diversity at vertices.

Contribution

It introduces a new condition linking local colour diversity to the existence of long properly coloured paths and cycles in edge-coloured graphs.

Findings

Existence of a properly coloured path of length 2d under the condition.

Existence of a properly coloured cycle of length at least d+1.

If no properly coloured cycle exists, a long properly coloured path of length 3×2^{d-1}-2 exists.

Abstract

Let $c$ be an edge-colouring of a graph $G$ such that for every vertex $v$ there are at least $d \ge 2$ different colours on edges incident to $v$. We prove that $G$ contains a properly coloured path of length 2d or a properly coloured cycle of length at least $d+1$. Moreover, if $G$ does not contain any properly coloured cycle, then there exists a properly coloured path of length $3 \times 2^{d-1}-2$.