The Hamilton cycle space of random regular graphs and randomly perturbed graphs
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909677182255104 |
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| author | Hefetz, Dan Krivelevich, Michael |
| author_facet | Hefetz, Dan Krivelevich, Michael |
| contents | 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 $δn$ for some constant $δ> 0$, and $G \sim \mathbb{G}(n, C/n)$, where $C := C(δ)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_04488 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Hamilton cycle space of random regular graphs and randomly perturbed graphs Hefetz, Dan Krivelevich, Michael Combinatorics 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 $δn$ for some constant $δ> 0$, and $G \sim \mathbb{G}(n, C/n)$, where $C := C(δ)$ 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. |
| title | The Hamilton cycle space of random regular graphs and randomly perturbed graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2507.04488 |