Colorful Hamilton cycles in random graphs

TL;DR

This paper investigates the set of possible color distributions in Hamilton cycles within randomly edge-colored graphs, especially near the threshold for Hamilton cycle existence, revealing how color profiles behave in such random settings.

Contribution

It introduces the concept of Hamilton cycle color profile in randomly colored graphs and analyzes its properties near critical thresholds for Hamilton cycle existence.

Findings

Characterizes the color profile set near the Hamilton cycle threshold.

Identifies conditions for the full color profile set to appear.

Provides probabilistic thresholds for the emergence of specific color distributions.

Abstract

Given an $n$ vertex graph whose edges have colored from one of $r$ colors $C=\{c_1,c_2,\ldots,c_r\}$, we define the Hamilton cycle color profile $hcp(G)$ to be the set of vectors $(m_1,m_2,\ldots,m_r)\in [0,n]^r$ such that there exists a Hamilton cycle that is the concatenation of $r$ paths $P_1,P_2,\ldots,P_r$, where $P_i$ contains $m_i$ edges of color $c_i$. We study $hcp(G_{n,p})$ when the edges are randomly colored. We discuss the profile close to the threshold for the existence of a Hamilton cycle and the threshold for when $hcp(G_{n,p})=\{(m_1,m_2,\ldots,m_r)\in [0,n]^r: m_1+m_2+\cdots+m_r=n\}$.