The completion numbers of Hamiltonicity and pancyclicity in random graphs

TL;DR

This paper investigates the completion number μ(G) for Hamiltonicity and the analogous completion number hatμ(G) for pancyclicity in binomial random graphs G(n,p). It derives a sharp asymptotic μ(G) = (1+o(1)) f(d)·n for a wide range 20 ≤ d = np ≤ 0.4 log n, with f(d) given by an explicit expansion involving terms like (1/2) d e^{−d} and higher-order contributions e^{−3d}; the authors also show in denser regimes that μ(G) equals a simple degree-based lower bound n_0(G) + ⌈n_1(G)/2⌉. Moreover, hatμ(G) is shown to equal μ(G) whp for np ≥ 20, and a polynomial-time algorithm is provided to compute a minimal completing set achieving pancyclicity. The analysis leverages the strong 4-core structure and a detailed local-to-global decomposition via disjoint path covers, connecting combinatorial, probabilistic, and algorithmic techniques to characterize completion thresholds and hitting times in the random-graph process. Overall, the work unifies structural graph theory with probabilistic methods to quantify how many edges are needed to enforce Hamiltonicity and pancyclicity in random graphs, with implications for efficient graph augmentation in random-like networks.

Abstract

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.

Figures (7)

• Figure 1: A $3$-spider in $G$ with vertex set $\{a,b,c,d,u,v,w\}$. The vertices $u,v,w$ have degree $2$ and a common neighbour, and the edges incident to them span a tree.
• Figure 2: The disjoint path covering $P$, all of whose endpoints (except for maybe one) lie in $A(G)$. The dotted edges represent the edges of the matching $M$, and the snaked lines are the paths of $Q$.
• Figure 3: Modifying the covering of $C_k$ in $Q$ to obtain $Q^*$, and adding $\{a_k^k,a_1^k\}$ to $F_0$. Adding $F_0$ to $G$ will now result in a graph that contains a $k$-cycle.
• Figure 4: The set $F_1$ (dashed), which corresponds to the matching $M$ (dotted). Adding $F_1$ to the graph incorporates the paths of $Q^*$ with one endpoint in $A(G)$ and one in $B(G)$ (snaked) into a long path that covers $C(G)$ and all the vertices in $B(G)$ that are not internal in $Q^*$, starting at $x_1$ and ending at $x_2$.
• Figure 5: With high probability, $G\cup F$ contains a cycle $H_{\ell}$ that passes through all the vertices except for $u(X_{\ell+1}),u(X_{\ell+2}),...,u(X_{s})$, and therefore a cycle of length $n-(\ell -s)$, for all $0\le \ell \le s$.

Theorems & Definitions (51)

• Definition 1
• Theorem 1
• Definition 2
• Theorem 2
• Corollary 1
• proof
• Corollary 2
• Theorem 3
• Theorem 4
• Lemma 2.1: Asymptotic equivalence of $G(n,p)$ and $G_t$, see e.g. FK, Lemma 1.2