Packing Hamilton Cycles in Cores of Random Graphs

TL;DR

This paper proves that the $k$-core of a random graph process contains multiple edge-disjoint Hamilton cycles and a large matching, confirming conjectures and improving previous results for $k ext{-cores}$ with $k ext{ } extgreater 3$.

Contribution

It establishes the presence of multiple Hamilton cycles and a large matching in the $k$-core of random graphs, confirming conjectures and extending known results.

Findings

For odd $k$, $G_{t}^{(k)}$ contains $(k-3)/2$ Hamilton cycles plus a 2-factor.

For even $k$, $G_{t}^{(k)}$ contains $(k-2)/2$ Hamilton cycles plus a large matching.

$G_{t}^{(k)}$ is Hamiltonian for $k extgreater 3$ and $t extgreater au_k$.

Abstract

Consider the random graph process $\{G_t\}_{t\geq 0}$. For $k\geq 3$ let $G_{t}^{(k)}$ denote the $k$-core of $G_t$ and let $\tau_k$ be the minimum $t$ such that the $k$-core of $G_t$ is nonempty. It is well known that w.h.p. for $G_{\tau_k}^{(k)}$ has linear size while it is believed to be Hamiltonian. Bollob\'{a}s, Cooper, Fenner and Frieze further conjectured that w.h.p. $G_{t}^{(k)}$ spans $\lfloor \frac{k-1}{2} \rfloor$ edge-disjoint Hamilton cycles plus, when $k$ is even, a perfect matching for $t\geq \tau_k$. We prove that w.h.p.\@ if $k$ is odd then $G_{t}^{(k)}$ spans $\frac{k-3}{2}$ edge disjoint Hamilton cycles plus an additional 2-factor whereas if $k$ is even then it spans $\frac{k-2}{2}$ edge disjoint Hamilton cycles plus an additional matching of size $n/2-o(n)$ for $t\geq \tau_k$. In particular w.h.p. $G_{t}^{(k)}$ is Hamiltonian for $k\geq 4$ and $t\geq \tau_k$. This improves upon results of Krivelevich, Lubetzky and Sudakov.