The completion numbers of Hamiltonicity and pancyclicity in random graphs

Abstract

Let $\mu(G)$ denote the minimum number of edges whose addition to $G$ results in a Hamiltonian graph, and let $\hat{\mu}(G)$ denote the minimum number of edges whose addition to $G$ results in a pancyclic graph. We study the distributions of $\mu (G),\hat{\mu}(G)$ 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 $\mu (G)= (1+o(1))\cdot f(d)\cdot n$. Let $n_i(G)$ denote the number of degree $i$ vertices in $G$. A trivial lower bound on $\mu(G)$ 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, $\mu (G) = n_0(G) + \lceil \frac{1}{2}n_1(G) \rceil$. For completion to pancyclicity, we show that if $G\sim G(n,p)$ and $np\ge 20$ then, with high probability, $\hat{\mu} (G)=\mu (G)$. 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 $\mu (G)$ whose addition to $G$ results in a pancyclic (and therefore also Hamiltonian) graph.