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arXiv · 2103.03916

Colorful Hamilton cycles in random graphs

Debsoumya Chakraborti·Alan Frieze·Mihir Hasabnis
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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 $hcp(G_{n,p})=\{(m_1,m_2,\ldots,m_r)\in [0,n]^r: m_1+m_2+\cdots+m_r=n\}$.

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Debsoumya Chakraborti, Alan Frieze, Mihir Hasabnis. 2022-10-05. Colorful Hamilton cycles in random graphs. https://doi.org/10.1137/21m1403291

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