Hamiltonian Complete Number of Some Variants of Caterpillar Graphs

TL;DR

This paper studies the minimum number of edges needed to add to certain caterpillar graphs to make them Hamiltonian, providing explicit formulas for regular cases and bounds for irregular ones, advancing understanding of Hamiltonian properties in trees.

Contribution

It derives explicit formulas and bounds for the Hamiltonian complete number in both regular and irregular caterpillar graphs, a novel analysis in this graph class.

Findings

For regular caterpillar graphs, λ_H(G) = n(k-1).

Bounds are provided for irregular caterpillar graphs.

Results enhance understanding of Hamiltonian properties in tree-like structures.

Abstract

A graph $G$ is said to be Hamiltonian if it contains a spanning cycle. In this work, we investigate the Hamiltonian completeness of certain classes of caterpillar graphs, which are trees with a central path to which all other vertices are adjacent. For a non-Hamiltonian graph $G$, the Hamiltonian complete number $\lambda_H(G)$ is the minimum number of edges that must be added to $G$ to make it Hamiltonian. We focus on both regular and irregular caterpillar graphs, deriving explicit formulas for $\lambda_H(G)$ in various cases. Specifically, we show that for a regular caterpillar graph $G_{n(k)}$ where each vertex on the central path is adjacent to $k$ leaves, $\lambda_H(G_{n(k)}) = n(k-1)$. We also explore irregular caterpillar graphs, where the number of leaves adjacent to each vertex on the central path varies, and provide bounds for $\lambda_H(G)$ in these cases. Our results contribute to the understanding of Hamiltonian properties in tree-like structures and have potential applications in network design and optimization.