Hamiltonicity of the Complete Double Vertex Graph of some Join Graphs

TL;DR

This paper investigates the Hamiltonian properties of the complete double vertex graph of certain join graphs, providing an infinite family of graphs where the double vertex graph is Hamiltonian, regardless of the original graph's Hamiltonicity.

Contribution

It introduces an infinite family of graphs G for which the complete double vertex graph M_2(G) is Hamiltonian, expanding understanding of Hamiltonicity in derived graph structures.

Findings

M_2(G) is Hamiltonian for an infinite family of graphs G.

Includes graphs G that are both Hamiltonian and non-Hamiltonian.

Provides conditions under which M_2(G) maintains Hamiltonicity.

Abstract

The complete double vertex graph $M_2(G)$ of $G$ is defined as the graph whose vertices are the $2$-multisubsets of $V(G)$, and two of such vertices are adjacent in $M_2(G)$ if their symmetric difference (as multisets) is a pair of adjacent vertices in $G$. In this paper we exhibit an infinite family of graphs $G$ (containing Hamiltonian and non-Hamiltonian graphs) for which $M_2(G)$ are Hamiltonian.