The Completion Numbers of Hamiltonicity and Pancyclicity in Random Graphs - Alon, Yahav; Anastos, Michael (Random structures & algorithms, 2025)
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.