Hamiltonian Complete Number of Some Variants of Caterpillar Graphs

TL;DR

This work investigates the Hamiltonian complete number $\lambda_H(G)$, the minimum number of edges needed to make a non-Hamiltonian graph $G$ Hamiltonian, with a focus on caterpillar graphs. For regular caterpillars $G_{n(k)}$ on a central path $P_n$ with $k$ leaves per path vertex, it derives exact formulas: $\lambda_H(G_{n(1)}) = \lceil n/2 \rceil$, $\lambda_H(G_{n(2)}) = n$, and, in general, $\lambda_H(G_{n(k)}) = n(k-1)$. The irregular case is shown to depend on leaf distribution along the central path, yielding bounds and exact results in special cases (notably $\lambda_H(G) = \sum_{i=1}^n l(v_i) - n$ when $l(v_i) \ge 3$ for all $i$). The paper introduces constructs such as $(0,1)$-leaf segment paths and deserted pendants to analyze irregular caterpillars, advancing understanding of Hamiltonicity in tree-like graphs and informing potential network-design applications.

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 $λ_H(G)$ 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 $λ_H(G)$ 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, $λ_H(G_{n(k)}) = n(k-1)$. We also explore irregular caterpillar graphs, where the number of leaves adjacent to each vertex on the central path varies, and provide bounds for $λ_H(G)$ 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.

Figures (8)

• Figure 1: Caterpillar Graph $G_5$
• Figure 2: $G_5$ in Figure \ref{['fig1']} has transformed into an Hamiltonian graph by three extra edges, showing that $\lambda_H(G_5) = 3.$
• Figure 3: Caterpillar Graph $G_4$
• Figure 4: $G_4$ in Figure \ref{['fig3']} has transformed into an Hamiltonian graph by two extra edges, showing that $\lambda_H(G_4) = 2.$
• Figure 5: A Caterpillar Graph $G_{5(2)}$ with central path of 5 vertices, where each vertex is connected to two leaves.

Theorems & Definitions (34)

• Theorem 2.1
• Theorem 2.2
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.1
• Definition 3.1
• Definition 3.2
• Remark 3.2