Cores of random graphs are born Hamiltonian

TL;DR

This paper proves that for sufficiently large fixed k, the k-core of a random graph process is Hamiltonian immediately upon emergence and contains multiple edge-disjoint Hamilton cycles, confirming long-standing conjectures.

Contribution

It establishes the Hamiltonicity of the k-core for large fixed k immediately after its formation and confirms the existence of multiple edge-disjoint Hamilton cycles.

Findings

k-core is Hamiltonian immediately after emergence for large fixed k

Contains multiple edge-disjoint Hamilton cycles for large fixed k

Confirms conjectures about Hamiltonicity and cycle packing in k-cores

Abstract

Let $(G_t)_{t \geq 0}$ be the random graph process ($G_0$ is edgeless and $G_t$ is obtained by adding a uniformly distributed new edge to $G_{t-1}$), and let $\tau_k$ denote the minimum time $t$ such that the $k$-core of $G_t$ (its unique maximal subgraph with minimum degree at least $k$) is nonempty. For any fixed $k\geq 3$ the $k$-core is known to emerge via a discontinuous phase transition, where at time $t=\tau_k$ its size jumps from 0 to linear in the number of vertices with high probability. It is believed that for any $k\geq 3$ the core is Hamiltonian upon creation w.h.p., and Bollob\'as, Cooper, Fenner and Frieze further conjectured that it in fact admits $\lfloor(k-1)/2\rfloor$ edge-disjoint Hamilton cycles. However, even the asymptotic threshold for Hamiltonicity of the $k$-core in $G(n,p)$ was unknown for any $k$. We show here that for any fixed $k\ge 15$ the $k$-core of $G_t$ is w.h.p. Hamiltonian for all $t \geq \tau_k$, i.e., immediately as the $k$-core appears and indefinitely afterwards. Moreover, we prove that for large enough fixed $k$ the $k$-core contains $\lfloor (k-3)/2\rfloor$ edge-disjoint Hamilton cycles w.h.p. for all $t\geq \tau_k$.