The completion numbers of Hamiltonicity and pancyclicity in random graphs
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2023
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| _version_ | 1866913659580579840 |
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| author | Alon, Yahav Anastos, Michael |
| author_facet | Alon, Yahav Anastos, Michael |
| contents | Let $μ(G)$ denote the minimum number of edges whose addition to $G$ results in a Hamiltonian graph, and let $\hatμ(G)$ denote the minimum number of edges whose addition to $G$ results in a pancyclic graph. We study the distributions of $μ(G),\hatμ(G)$ in the context of binomial random graphs.
Letting $d=d(n) := n\cdot p$, we prove that there exists a function $f:\mathbb{R}^+\to [0,1]$ of order $f(d) = \frac{1}{2}de^{-d}+e^{-d}+O(d^6e^{-3d})$ such that, if $G\sim G(n,p)$ with $20 \le d(n) \le 0.4 \log n$, then with high probability $μ(G)= (1+o(1))\cdot f(d)\cdot n$.
Let $n_i(G)$ denote the number of degree $i$ vertices in $G$. A trivial lower bound on $μ(G)$ is given by the expression $n_0(G) + \lceil \frac{1}{2}n_1(G) \rceil$. In the denser regime of random graphs, we show that if $np-\frac{1}{3}\log n - 2\log \log n \to \infty$ and $G\sim G(n,p)$ then, with high probability, $μ(G) = n_0(G) + \lceil \frac{1}{2}n_1(G) \rceil$.
For completion to pancyclicity, we show that if $G\sim G(n,p)$ and $np\ge 20$ then, with high probability, $\hatμ (G)=μ(G)$.
Finally, we present a polynomial time algorithm such that, if $G\sim G(n,p)$ and $np\ge 20$, then, with high probability, the algorithm returns a set of edges of size $μ(G)$ whose addition to $G$ results in a pancyclic (and therefore also Hamiltonian) graph. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_03710 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The completion numbers of Hamiltonicity and pancyclicity in random graphs Alon, Yahav Anastos, Michael Combinatorics Let $μ(G)$ denote the minimum number of edges whose addition to $G$ results in a Hamiltonian graph, and let $\hatμ(G)$ denote the minimum number of edges whose addition to $G$ results in a pancyclic graph. We study the distributions of $μ(G),\hatμ(G)$ in the context of binomial random graphs. Letting $d=d(n) := n\cdot p$, we prove that there exists a function $f:\mathbb{R}^+\to [0,1]$ of order $f(d) = \frac{1}{2}de^{-d}+e^{-d}+O(d^6e^{-3d})$ such that, if $G\sim G(n,p)$ with $20 \le d(n) \le 0.4 \log n$, then with high probability $μ(G)= (1+o(1))\cdot f(d)\cdot n$. Let $n_i(G)$ denote the number of degree $i$ vertices in $G$. A trivial lower bound on $μ(G)$ is given by the expression $n_0(G) + \lceil \frac{1}{2}n_1(G) \rceil$. In the denser regime of random graphs, we show that if $np-\frac{1}{3}\log n - 2\log \log n \to \infty$ and $G\sim G(n,p)$ then, with high probability, $μ(G) = n_0(G) + \lceil \frac{1}{2}n_1(G) \rceil$. For completion to pancyclicity, we show that if $G\sim G(n,p)$ and $np\ge 20$ then, with high probability, $\hatμ (G)=μ(G)$. Finally, we present a polynomial time algorithm such that, if $G\sim G(n,p)$ and $np\ge 20$, then, with high probability, the algorithm returns a set of edges of size $μ(G)$ whose addition to $G$ results in a pancyclic (and therefore also Hamiltonian) graph. |
| title | The completion numbers of Hamiltonicity and pancyclicity in random graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2304.03710 |