The completion numbers of Hamiltonicity and pancyclicity in random graphs

TL;DR

This paper investigates the minimum number of edges needed to make a random graph Hamiltonian or pancyclic, providing probabilistic bounds, exact formulas in dense regimes, and an efficient algorithm for achieving pancyclicity.

Contribution

It introduces new probabilistic bounds and formulas for the completion numbers of Hamiltonicity and pancyclicity in random graphs, along with a polynomial-time algorithm for constructing pancyclic graphs.

Findings

High probability bounds for (G) and (G) in various regimes

Exact formula for (G) in dense random graphs

Efficient algorithm for creating pancyclic 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.