The Pairing-Hamiltonian property in graph prisms

TL;DR

The paper investigates the problem of extending perfect matchings to Hamiltonian cycles via the Pairing-Hamiltonian property (PH-property). Building on Fink's theorem that the hypercube $\mathcal{Q}_d$ has PH for $d\ge2$, it proves that if a graph $G$ has PH, then its prism $\mathcal{P}(G)=G\Box K_2$ and all iterates $\mathcal{P}^k(G)$ inherit the PH-property. It further establishes a convergence phenomenon: for any connected $G$, a sufficiently large prism power $\mathcal{P}^k(G)$ has PH, with explicit bounds in terms of the minimum leaf number $\mathrm{ml}(G)$ and a special case when $G$ is traceable, implying $\mathcal{P}^{m+3}(G)$ has PH with $m=\mathrm{ml}(G)$. These results generalize Kreweras's conjecture and provide a framework for generating PH-graphs through prism operations, while outlining several open problems for broader product-compatibility and faster convergence.

Abstract

Let $G$ be a graph of even order, and consider $K_G$ as the complete graph on the same vertex set as $G$. A perfect matching of $K_G$ is called a pairing of $G$. If for every pairing $M$ of $G$ it is possible to find a perfect matching $N$ of $G$ such that $M \cup N$ is a Hamiltonian cycle of $K_G$, then $G$ is said to have the Pairing-Hamiltonian property, or PH-property, for short. In 2007, Fink [J. Combin. Theory Ser. B, 97] proved that for every $d\geq 2$, the $d$-dimensional hypercube $\mathcal{Q}_d$ has the PH-property, thus proving a conjecture posed by Kreweras in 1996. In this paper we extend Fink's result by proving that given a graph $G$ having the PH-property, the prism graph $\mathcal{P}(G)$ of $G$ has the PH-property as well. Moreover, if $G$ is a connected graph, we show that there exists a positive integer $k_0$ such that the $k^{\textrm{th}}$-prism of a graph $\mathcal{P}^k(G)$ has the PH-property for all $k \ge k_0$.

Figures (2)

• Figure 1: An extension of the pairing $P$, depicted in bold, when $|X|=0$. The dashed edges represent those in $N$, whereas the dotted edges are the edges $e_i$ and $e'_i$.
• Figure 2: An extension of the pairing $P$, depicted in bold, when $|X|= 2r>0$. The dashed edges represent those in $M$ and $M'$, whereas the dotted edges are those in $L$ and $R$.

Theorems & Definitions (13)

• Theorem 1.1: Fink, Fink 2007
• Theorem 1.2: Alahmadi et al., 2015 AAAHST
• Theorem 2.1
• proof
• Corollary 3.1
• Definition 3.2
• Lemma 3.3
• proof
• Proposition 3.4
• proof