An extension of a cubic 2-connected plane graph G to a hamiltonian plane graph contained in G^{2}

TL;DR

This paper demonstrates how to extend a simple cubic 2-connected plane graph into a Hamiltonian plane graph within its square, maintaining certain degree constraints and relating the number of edges to the original graph's components.

Contribution

It introduces a method to extend cubic 2-connected plane graphs into Hamiltonian graphs within their squares, with specific edge and degree properties, expanding understanding of graph extensions.

Findings

Existence of Hamiltonian extensions within the square of G

Edge count relation: |E(J)|=|E(G)|+2n-2

Maximum degree of J is at most 5

Abstract

Let $G$ be a simple cubic 2-connected plane graph. For every $2$-factor $X$ of $G$ having $n$-components there exists a simple hamiltonian plane graph $J \subset G^{2}$ such that $|E(J)|= |E(G)| + 2n -2$ and $\Delta(J) \leqslant 5$.