The Hamilton cycle space of random regular graphs and randomly perturbed graphs

TL;DR

This paper investigates the cycle space of random regular graphs and perturbed graphs, establishing conditions under which the cycle space is spanned by Hamilton cycles, extending known Hamiltonicity results.

Contribution

It proves that for large degree, the cycle space of random regular graphs is generated by Hamilton cycles, and extends Hamiltonicity results to the cycle space of perturbed graphs.

Findings

Cycle space equals the span of Hamilton cycles for large degree graphs.

Asymptotic almost sure equality of cycle space and Hamilton cycle span in random regular graphs.

Strengthening of Hamiltonicity results in perturbed graphs with high minimum degree.

Abstract

The cycle space of a graph $G$, denoted $C(G)$, is a vector space over ${\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $C_n(G)$ is the subspace of $C(G)$, spanned by the incidence vectors of Hamilton cycles of $G$. We prove that asymptotically almost surely $C_n(G_{n,d}) = C(G_{n,d})$ holds whenever $n$ is odd and $d$ is a sufficiently large (even) integer. This extends (though with a weaker bound on $d$) the well-known result asserting that $G_{n,d}$ is asymptotically almost surely Hamiltonian for every $d \geq 3$ (but not for $d < 3$). Since $n$ being odd mandates that $d$ be even, somewhat limiting the generality of our result, we also prove that if $n$ is even and $d$ is any sufficiently large integer, then asymptotically almost surely $C_{n-1}(G_{n,d}) = C(G_{n,d})$. An influential result of Bohman, Frieze, and Martin asserts that if $H$ is an $n$-vertex graph with minimum degree at least $\delta n$ for some constant $\delta > 0$, and $G \sim \mathbb{G}(n, C/n)$, where $C := C(\delta)$ is a sufficiently large constant, then $H \cup G$ is asymptotically almost surely Hamiltonian. We strengthen this result by proving that the same assumptions on $H$ and $G$ ensure that $C_n(H \cup G) = C(H \cup G)$ holds asymptotically almost surely.