Hamiltonian paths in iterated line graphs

TL;DR

The paper investigates Hamiltonian paths in iterated line graphs by introducing the Hamiltonian path index $h_p(G)$, the minimal iteration $n$ for which $L^n(G)$ contains a Hamiltonian path. It proves the existence of $h_p(G)$ for all connected graphs and derives an exact formula for trees in terms of endpaths and a branch-length parameter $k(b)$, using the interplay between endpaths, dominating trails, and the block-chain structure of iterated line graphs. The proofs combine domination-trail criteria with a detailed analysis of $L^m(T)$'s block structure to both construct Hamiltonian paths and establish tight lower bounds. These results deepen understanding of how iterated line graphs acquire Hamiltonian paths, provide precise criteria for trees, and lay groundwork for extending similar analysis to broader graph classes.

Abstract

For integer $n$, the $n$-iterated line graph $L^n(G)$ of an undirected graph $G$ is defined to be $L(L^{n-1}(G))$, where $L^1(G)$ is the line graph $L(G)$ of $G$. In this paper we introduce hamiltonian path index. Hamiltonian path index, denoted by $h_p(G)$, is the minimum number $n$ such that $L^n(G)$ contains a hamiltonian path. We show that hamiltonian path index of $G$ exists for any graph $G$ and we set the exact value of hamiltonian path index for trees and discuss the problem about graphs with hamiltonian 2-connected blocks.

Figures (1)

• Figure 1: The $v_1v$-endpath $P_1$ in a tree $T$ (bold edges).

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• Theorem 3
• proof