The Completion Numbers of Hamiltonicity and Pancyclicity in Random Graphs
ABSTRACT Let μ(G)$$ \mu (G) $$ denote the minimum number of edges whose addition to G$$ G $$ results in a Hamiltonian graph, and let μ^(G)$$ \hat{\mu}(G) $$ denote the minimum number of edges whose addition to G$$ G $$ results in a pancyclic graph. We study the distributions of μ(G),μ^(G)$$ \mu (G),...
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| Vydáno v: | Random structures & algorithms Ročník 66; číslo 2 |
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John Wiley & Sons, Inc
01.03.2025
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| Abstract | ABSTRACT
Let μ(G)$$ \mu (G) $$ denote the minimum number of edges whose addition to G$$ G $$ results in a Hamiltonian graph, and let μ^(G)$$ \hat{\mu}(G) $$ denote the minimum number of edges whose addition to G$$ G $$ results in a pancyclic graph. We study the distributions of μ(G),μ^(G)$$ \mu (G),\hat{\mu}(G) $$ in the context of binomial random graphs. Letting d=d(n):=n·p$$ d=d(n):= n\cdotp p $$, we prove that there exists a function f:ℝ+→[0,1]$$ f:{\mathbb{R}}^{+}\to \left[0,1\right] $$ of order f(d)=12de−d+e−d+O(d6e−3d)$$ f(d)=\frac{1}{2}d{e}^{-d}+{e}^{-d}+O\left({d}^6{e}^{-3d}\right) $$ such that, if G∼G(n,p)$$ G\sim G\left(n,p\right) $$ with 20≤d(n)≤0.4logn$$ 20\le d(n)\le 0.4\log n $$, then with high probability μ(G)=(1+o(1))·f(d)·n$$ \mu (G)=\left(1+o(1)\right)\cdotp f(d)\cdotp n $$. Let ni(G)$$ {n}_i(G) $$ denote the number of degree i$$ i $$ vertices in G$$ G $$. A trivial lower bound on μ(G)$$ \mu (G) $$ is given by the expression n0(G)+⌈12n1(G)⌉$$ {n}_0(G)+\left\lceil \frac{1}{2}{n}_1(G)\right\rceil $$. We show that in the random graph process {Gt}t=0n2$$ {\left\{{G}_t\right\}}_{t=0}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)} $$ with high probability there exist times t1,t2$$ {t}_1,{t}_2 $$, both of order (1+o(1))n·16logn+loglogn$$ \left(1+o(1)\right)n\cdotp \left(\frac{1}{6}\log n+\mathrm{loglog}n\right) $$, such that μ(Gt)=n0(Gt)+⌈12n1(Gt)⌉$$ \mu \left({G}_t\right)={n}_0\left({G}_t\right)+\left\lceil \frac{1}{2}{n}_1\left({G}_t\right)\right\rceil $$ for every t≥t1$$ t\ge {t}_1 $$ and μ(Gt)>n0(Gt)+⌈12n1(Gt)⌉$$ \mu \left({G}_t\right)>{n}_0\left({G}_t\right)+\left\lceil \frac{1}{2}{n}_1\left({G}_t\right)\right\rceil $$ for every t≤t2$$ t\le {t}_2 $$. The time ti$$ {t}_i $$ can be characterized as the smallest t$$ t $$ for which Gt$$ {G}_t $$ contains less than i$$ i $$ copies of a certain problematic subgraph. In particular, this implies that the hitting time for the existence of a Hamilton path is equal to the hitting time of n1(G)≤2$$ {n}_1(G)\le 2 $$ with high probability. For the binomial random graph, this implies that if np−13logn−2loglogn→∞$$ np-\frac{1}{3}\log n-2\mathrm{loglog}n\to \infty $$ and G∼G(n,p)$$ G\sim G\left(n,p\right) $$, then, with high probability, μ(G)=n0(G)+⌈12n1(G)⌉$$ \mu (G)={n}_0(G)+\left\lceil \frac{1}{2}{n}_1(G)\right\rceil $$. For completion to pancyclicity, we show that if G∼G(n,p)$$ G\sim G\left(n,p\right) $$ and np≥20$$ np\ge 20 $$, then, with high probability, μ^(G)=μ(G)$$ \hat{\mu}(G)=\mu (G) $$. Finally, we present a polynomial time algorithm such that, if G∼G(n,p)$$ G\sim G\left(n,p\right) $$ and np≥20$$ np\ge 20 $$, then, with high probability, the algorithm returns a set of edges of size μ(G)$$ \mu (G) $$ whose addition to G$$ G $$ results in a pancyclic (and therefore also Hamiltonian) graph. |
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| AbstractList | Let denote the minimum number of edges whose addition to results in a Hamiltonian graph, and let denote the minimum number of edges whose addition to results in a pancyclic graph. We study the distributions of in the context of binomial random graphs. Letting , we prove that there exists a function of order such that, if with , then with high probability . Let denote the number of degree vertices in . A trivial lower bound on is given by the expression . We show that in the random graph process with high probability there exist times , both of order , such that for every and for every . The time can be characterized as the smallest for which contains less than copies of a certain problematic subgraph. In particular, this implies that the hitting time for the existence of a Hamilton path is equal to the hitting time of with high probability. For the binomial random graph, this implies that if and , then, with high probability, . For completion to pancyclicity, we show that if and , then, with high probability, . Finally, we present a polynomial time algorithm such that, if and , then, with high probability, the algorithm returns a set of edges of size whose addition to results in a pancyclic (and therefore also Hamiltonian) graph. Let μ(G)$$ \mu (G) $$ denote the minimum number of edges whose addition to G$$ G $$ results in a Hamiltonian graph, and let μ^(G)$$ \hat{\mu}(G) $$ denote the minimum number of edges whose addition to G$$ G $$ results in a pancyclic graph. We study the distributions of μ(G),μ^(G)$$ \mu (G),\hat{\mu}(G) $$ in the context of binomial random graphs. Letting d=d(n):=n·p$$ d=d(n):= n\cdotp p $$, we prove that there exists a function f:ℝ+→[0,1]$$ f:{\mathbb{R}}^{+}\to \left[0,1\right] $$ of order f(d)=12de−d+e−d+O(d6e−3d)$$ f(d)=\frac{1}{2}d{e}^{-d}+{e}^{-d}+O\left({d}^6{e}^{-3d}\right) $$ such that, if G∼G(n,p)$$ G\sim G\left(n,p\right) $$ with 20≤d(n)≤0.4logn$$ 20\le d(n)\le 0.4\log n $$, then with high probability μ(G)=(1+o(1))·f(d)·n$$ \mu (G)=\left(1+o(1)\right)\cdotp f(d)\cdotp n $$. Let ni(G)$$ {n}_i(G) $$ denote the number of degree i$$ i $$ vertices in G$$ G $$. A trivial lower bound on μ(G)$$ \mu (G) $$ is given by the expression n0(G)+⌈12n1(G)⌉$$ {n}_0(G)+\left\lceil \frac{1}{2}{n}_1(G)\right\rceil $$. We show that in the random graph process {Gt}t=0n2$$ {\left\{{G}_t\right\}}_{t=0}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)} $$ with high probability there exist times t1,t2$$ {t}_1,{t}_2 $$, both of order (1+o(1))n·16logn+loglogn$$ \left(1+o(1)\right)n\cdotp \left(\frac{1}{6}\log n+\mathrm{loglog}n\right) $$, such that μ(Gt)=n0(Gt)+⌈12n1(Gt)⌉$$ \mu \left({G}_t\right)={n}_0\left({G}_t\right)+\left\lceil \frac{1}{2}{n}_1\left({G}_t\right)\right\rceil $$ for every t≥t1$$ t\ge {t}_1 $$ and μ(Gt)>n0(Gt)+⌈12n1(Gt)⌉$$ \mu \left({G}_t\right)>{n}_0\left({G}_t\right)+\left\lceil \frac{1}{2}{n}_1\left({G}_t\right)\right\rceil $$ for every t≤t2$$ t\le {t}_2 $$. The time ti$$ {t}_i $$ can be characterized as the smallest t$$ t $$ for which Gt$$ {G}_t $$ contains less than i$$ i $$ copies of a certain problematic subgraph. In particular, this implies that the hitting time for the existence of a Hamilton path is equal to the hitting time of n1(G)≤2$$ {n}_1(G)\le 2 $$ with high probability. For the binomial random graph, this implies that if np−13logn−2loglogn→∞$$ np-\frac{1}{3}\log n-2\mathrm{loglog}n\to \infty $$ and G∼G(n,p)$$ G\sim G\left(n,p\right) $$, then, with high probability, μ(G)=n0(G)+⌈12n1(G)⌉$$ \mu (G)={n}_0(G)+\left\lceil \frac{1}{2}{n}_1(G)\right\rceil $$. For completion to pancyclicity, we show that if G∼G(n,p)$$ G\sim G\left(n,p\right) $$ and np≥20$$ np\ge 20 $$, then, with high probability, μ^(G)=μ(G)$$ \hat{\mu}(G)=\mu (G) $$. Finally, we present a polynomial time algorithm such that, if G∼G(n,p)$$ G\sim G\left(n,p\right) $$ and np≥20$$ np\ge 20 $$, then, with high probability, the algorithm returns a set of edges of size μ(G)$$ \mu (G) $$ whose addition to G$$ G $$ results in a pancyclic (and therefore also Hamiltonian) graph. ABSTRACT Let μ(G)$$ \mu (G) $$ denote the minimum number of edges whose addition to G$$ G $$ results in a Hamiltonian graph, and let μ^(G)$$ \hat{\mu}(G) $$ denote the minimum number of edges whose addition to G$$ G $$ results in a pancyclic graph. We study the distributions of μ(G),μ^(G)$$ \mu (G),\hat{\mu}(G) $$ in the context of binomial random graphs. Letting d=d(n):=n·p$$ d=d(n):= n\cdotp p $$, we prove that there exists a function f:ℝ+→[0,1]$$ f:{\mathbb{R}}^{+}\to \left[0,1\right] $$ of order f(d)=12de−d+e−d+O(d6e−3d)$$ f(d)=\frac{1}{2}d{e}^{-d}+{e}^{-d}+O\left({d}^6{e}^{-3d}\right) $$ such that, if G∼G(n,p)$$ G\sim G\left(n,p\right) $$ with 20≤d(n)≤0.4logn$$ 20\le d(n)\le 0.4\log n $$, then with high probability μ(G)=(1+o(1))·f(d)·n$$ \mu (G)=\left(1+o(1)\right)\cdotp f(d)\cdotp n $$. Let ni(G)$$ {n}_i(G) $$ denote the number of degree i$$ i $$ vertices in G$$ G $$. A trivial lower bound on μ(G)$$ \mu (G) $$ is given by the expression n0(G)+⌈12n1(G)⌉$$ {n}_0(G)+\left\lceil \frac{1}{2}{n}_1(G)\right\rceil $$. We show that in the random graph process {Gt}t=0n2$$ {\left\{{G}_t\right\}}_{t=0}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)} $$ with high probability there exist times t1,t2$$ {t}_1,{t}_2 $$, both of order (1+o(1))n·16logn+loglogn$$ \left(1+o(1)\right)n\cdotp \left(\frac{1}{6}\log n+\mathrm{loglog}n\right) $$, such that μ(Gt)=n0(Gt)+⌈12n1(Gt)⌉$$ \mu \left({G}_t\right)={n}_0\left({G}_t\right)+\left\lceil \frac{1}{2}{n}_1\left({G}_t\right)\right\rceil $$ for every t≥t1$$ t\ge {t}_1 $$ and μ(Gt)>n0(Gt)+⌈12n1(Gt)⌉$$ \mu \left({G}_t\right)>{n}_0\left({G}_t\right)+\left\lceil \frac{1}{2}{n}_1\left({G}_t\right)\right\rceil $$ for every t≤t2$$ t\le {t}_2 $$. The time ti$$ {t}_i $$ can be characterized as the smallest t$$ t $$ for which Gt$$ {G}_t $$ contains less than i$$ i $$ copies of a certain problematic subgraph. In particular, this implies that the hitting time for the existence of a Hamilton path is equal to the hitting time of n1(G)≤2$$ {n}_1(G)\le 2 $$ with high probability. For the binomial random graph, this implies that if np−13logn−2loglogn→∞$$ np-\frac{1}{3}\log n-2\mathrm{loglog}n\to \infty $$ and G∼G(n,p)$$ G\sim G\left(n,p\right) $$, then, with high probability, μ(G)=n0(G)+⌈12n1(G)⌉$$ \mu (G)={n}_0(G)+\left\lceil \frac{1}{2}{n}_1(G)\right\rceil $$. For completion to pancyclicity, we show that if G∼G(n,p)$$ G\sim G\left(n,p\right) $$ and np≥20$$ np\ge 20 $$, then, with high probability, μ^(G)=μ(G)$$ \hat{\mu}(G)=\mu (G) $$. Finally, we present a polynomial time algorithm such that, if G∼G(n,p)$$ G\sim G\left(n,p\right) $$ and np≥20$$ np\ge 20 $$, then, with high probability, the algorithm returns a set of edges of size μ(G)$$ \mu (G) $$ whose addition to G$$ G $$ results in a pancyclic (and therefore also Hamiltonian) graph. |
| Author | Alon, Yahav Anastos, Michael |
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| References | 2023; 30 2023 2022; 60 2021; 148 1991; 98 2022; 61 2024; 118 1987; 7 1987 1983; 43 1954; 6 1984 2016 2020; 57 2015 2016; 25 e_1_2_12_4_1 e_1_2_12_6_1 e_1_2_12_2_1 e_1_2_12_17_1 e_1_2_12_16_1 Cooper C. (e_1_2_12_5_1) 1987 Bollobás B. (e_1_2_12_3_1) 1984 e_1_2_12_15_1 e_1_2_12_14_1 e_1_2_12_13_1 e_1_2_12_12_1 e_1_2_12_8_1 e_1_2_12_11_1 e_1_2_12_7_1 e_1_2_12_10_1 e_1_2_12_9_1 |
| References_xml | – volume: 30 issue: 2 year: 2023 article-title: A Note on Long Cycles in Sparse Random Graphs publication-title: Electronic Journal of Combinatorics – start-page: 35 year: 1984 end-page: 57 – volume: 148 start-page: 184 year: 2021 end-page: 208 article-title: A Scaling Limit for the Length of the Longest Cycle in a Sparse Random Graph publication-title: Journal of Combinatorial Theory Series B – volume: 25 start-page: 269 year: 2016 end-page: 299 article-title: On the Method of Typical Bounded Differences publication-title: Combinatorics, Probability and Computing – volume: 7 start-page: 327 year: 1987 end-page: 341 article-title: An Algorithm for Finding Hamilton Paths and Cycles in Random Graphs publication-title: Combinatorica – volume: 61 start-page: 444 year: 2022 end-page: 461 article-title: Cycle Lengths in Sparse Random Graphs publication-title: Random Structures & Algorithms – year: 2023 – volume: 98 start-page: 231 year: 1991 end-page: 236 article-title: Cycles in Random Graphs publication-title: Discrete Mathematics – volume: 6 start-page: 347 year: 1954 end-page: 352 article-title: A Short Proof of the Factor Theorem for Finite Graphs publication-title: Canadian Journal of Mathematics – volume: 43 start-page: 55 year: 1983 end-page: 63 article-title: Limit Distributions for the Existence of Hamilton Circuits in a Random Graph publication-title: Discrete Mathematics – volume: 57 start-page: 32 issue: 1 year: 2020 end-page: 46 article-title: Finding a Hamilton Cycle Fast on Average Using Rotations and Extensions publication-title: Random Structures & Algorithms – volume: 60 start-page: 3 year: 2022 end-page: 24 article-title: A Scaling Limit for the Length of the Longest Cycle in a Sparse Random Digraph publication-title: Random Structures & Algorithms – start-page: 29 year: 1987 end-page: 39 – year: 2016 – volume: 118 year: 2024 article-title: Hamilton Completion and the Path Cover Number of Sparse Random Graphs publication-title: European Journal of Combinatorics – year: 2015 – ident: e_1_2_12_6_1 doi: 10.1007/978-3-319-31951-3 – ident: e_1_2_12_8_1 doi: 10.1137/1.9781611977554.ch88 – start-page: 35 volume-title: Graph Theory and Combinatorics year: 1984 ident: e_1_2_12_3_1 – ident: e_1_2_12_10_1 doi: 10.1017/CBO9781316339831 – ident: e_1_2_12_11_1 doi: 10.37236/11471 – ident: e_1_2_12_16_1 doi: 10.1002/rsa.21030 – ident: e_1_2_12_15_1 doi: 10.4153/CJM-1954-033-3 – ident: e_1_2_12_9_1 doi: 10.1007/BF02579321 – ident: e_1_2_12_14_1 doi: 10.1017/S0963548315000103 – ident: e_1_2_12_7_1 doi: 10.1002/rsa.20918 – ident: e_1_2_12_4_1 doi: 10.1016/j.ejc.2024.103934 – ident: e_1_2_12_13_1 doi: 10.1016/0012-365X(91)90379-G – ident: e_1_2_12_2_1 doi: 10.1016/0012-365X(83)90021-3 – ident: e_1_2_12_12_1 doi: 10.1016/j.jctb.2021.01.001 – start-page: 29 volume-title: Proceedings 3rd Annual Conference on Random Graphs year: 1987 ident: e_1_2_12_5_1 – ident: e_1_2_12_17_1 doi: 10.1002/rsa.21067 |
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| Snippet | ABSTRACT
Let μ(G)$$ \mu (G) $$ denote the minimum number of edges whose addition to G$$ G $$ results in a Hamiltonian graph, and let μ^(G)$$ \hat{\mu}(G) $$... Let denote the minimum number of edges whose addition to results in a Hamiltonian graph, and let denote the minimum number of edges whose addition to results... Let μ(G)$$ \mu (G) $$ denote the minimum number of edges whose addition to G$$ G $$ results in a Hamiltonian graph, and let μ^(G)$$ \hat{\mu}(G) $$ denote the... |
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| SubjectTerms | Algorithms Apexes Graph theory Graphs Hamilton cycles Lower bounds pancyclic graphs Polynomials random graph process random graphs |
| Title | The Completion Numbers of Hamiltonicity and Pancyclicity in Random Graphs |
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