Characterizing forbidden pairs for hamiltonian squares

TL;DR

This paper characterizes specific forbidden pairs in 4-connected graphs that determine the presence of a hamiltonian square, advancing understanding of graph structures related to hamiltonian cycles and their squares.

Contribution

It provides a complete characterization of forbidden pairs that guarantee the containment of a hamiltonian square in 4-connected graphs.

Findings

Identifies all forbidden pairs for hamiltonian squares in 4-connected graphs

Shows the necessity of 4-connectivity for the characterization

Highlights exceptions like K3 and K4 where the square is not 4-connected

Abstract

The square of a graph is obtained by adding additional edges joining all pair of vertices of distance two in the original graph. Particularly, if $C$ is a hamiltonian cycle of a graph $G$, then the square of $C$ is called a hamiltonian square of $G$. In this paper, we characterize all possible forbidden pairs, which implies the containment of a hamiltonian square, in a 4-connected graph. The connectivity condition is necessary as, except $K_3$ and $K_4$, the square of a cycle is always 4-connected.