Hamiltonicity of the Double Vertex Graph and the Complete Double Vertex Graph of some Join Graphs

Abstract

Let $G$ be a simple graph of order $n$. The double vertex graph $F_2(G)$ of $G$ is the graph whose vertices are the $2$-subsets of $V(G)$, where two vertices are adjacent in $F_2(G)$ if their symmetric difference is a pair of adjacent vertices in $G$. A generalization of this graph is the complete double vertex graph $M_2(G)$ of $G$, 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 (containing Hamiltonian and non-Hamiltonian graphs) for which $F_2(G)$ and $M_2(G)$ are Hamiltonian. This family of graphs is the set of join graphs $G=G_1+G_2$, where $G_1$ and $G_2$ are of order $m\ge 1$ and $n\ge 2$, respectively, and $G_2$ has a Hamiltonian path. For this family of graphs, we show that if $m\le 2n$ then $F_2(G)$ is Hamiltonian, and if $m\le 2(n-1)$ then $M_2(G)$ is Hamiltonian.

V2 is a revised version. Not intended for publication. Theorem 1.1 was presented in Symmetry 13(6) (1076), 2021, doi:10.3390/sym13061076 (arXiv:2101.01855) and Theorem 1.2 was presented in arXiv:2108.01119 (to appear in Matemática Contemporânea)

References in corpus (2)

• Constructing physically intuitive graph invariants
• On the Connectivity of Token Graphs of Trees

Cited by in corpus (1)

• Entanglement Entropy Bounds in the Higher Spin XXZ Chain