Long properly colored cycles in edge colored complete graphs

TL;DR

This paper improves bounds on the existence of long properly colored cycles in edge-colored complete graphs under certain maximum monochromatic degree conditions, advancing towards a conjecture about Hamiltonian cycles.

Contribution

It refines the minimum length bound for properly colored cycles in complete graphs with bounded monochromatic degree, moving closer to the conjectured Hamiltonian cycle result.

Findings

Improved the cycle length bound to rac{n}{2}rac{n}{2} + 2

Extended previous results from rac{n+2}{3}rac{n+2}{3} + 1

Progressed towards Bollobe1s and Erd
51s conjecture

Abstract

Let $K_{n}^{c}$ denote a complete graph on $n$ vertices whose edges are colored in an arbitrary way. Let $\Delta^{\mathrm{mon}} (K_{n}^{c})$ denote the maximum number of edges of the same color incident with a vertex of $K_{n}^{c}$. A properly colored cycle (path) in $K_{n}^{c}$ is a cycle (path) in which adjacent edges have distinct colors. B. Bollob\'{a}s and P. Erd\"{o}s (1976) proposed the following conjecture: if $\Delta^{\mathrm{mon}} (K_{n}^{c})<\lfloor \frac{n}{2} \rfloor$, then $K_{n}^{c}$ contains a properly colored Hamiltonian cycle. Li, Wang and Zhou proved that if $\Delta^{\mathrm{mon}} (K_{n}^{c})< \lfloor \frac{n}{2} \rfloor$, then $K_{n}^{c}$ contains a properly colored cycle of length at least $\lceil \frac{n+2}{3}\rceil+1$. In this paper, we improve the bound to $\lceil \frac{n}{2}\rceil + 2$.