Hamiltonian Cycles in the Square of a Graph

TL;DR

This paper establishes conditions under which the square of a graph, formed by vertex identification, remains Hamiltonian, and characterizes Hamiltonicity in specific block graphs with degree constraints.

Contribution

It introduces new necessary and sufficient conditions for the Hamiltonicity of the square of certain connected graphs with degree and block structure constraints.

Findings

Identifies conditions for Hamiltonian squares after vertex identification.

Proves Hamiltonicity criteria for block graphs with degree constraints.

Provides a characterization of Hamiltonian squares in specific graph classes.

Abstract

We show that under certain conditions the square of the graph obtained by identifying a vertex in two graphs with hamiltonian square is also hamiltonian. Using this result, we prove necessary and sufficient conditions for hamiltonicity of the square of a connected graph such that every vertex of degree at least three in a block graph corresponds to a cut vertex and any two these vertices are at distance at least four.