Hamiltonian properties in generalized lexicographic products

TL;DR

This paper investigates Hamiltonian properties in generalized lexicographic products of graphs, providing necessary and sufficient conditions for traceability and Hamiltonicity when the base graph is a path, extending previous results.

Contribution

It offers new criteria for Hamiltonian properties in generalized lexicographic products, broadening understanding beyond standard cases.

Findings

Established conditions for traceability in generalized lexicographic products.

Derived criteria for Hamiltonicity and Hamiltonian connectivity.

Extended prior results to more general graph replacements.

Abstract

The lexicographic product $G[H]$ of two graphs $G$ and $H$ is obtained from $G$ by replacing each vertex with a copy of $H$ and adding all edges between any pair of copies corresponding to adjacent vertices of $G$. We consider also the generalized lexicographic product such that we replace each vertex of $G$ with arbitrary graph on the same number of vertices. We present sufficient and necessary conditions for traceability, hamiltonicity and hamiltonian connectivity of $G[H]$ if $G$ is a path and hence we improved and extended results in M. Kriesell, A Note on Hamiltonian Cycles in Lexicographical Products.