# Some cyclic properties of $L_1$-graphs

## TL;DR

This paper investigates the cycle properties of $L_1$-graphs, showing that non-Hamiltonian cycles can be extended to larger cycles, and explores related path extension properties.

## Contribution

It extends previous work by demonstrating cycle extension properties in $L_1$-graphs, including non-Hamiltonian cycles and paths with specific endpoint conditions.

## Key findings

- Non-Hamiltonian cycles can be extended to larger cycles containing all original vertices.
- Not all $L_1$-graphs are pancyclic, contrary to some earlier assumptions.
- Cycle extension results also apply to certain paths with non-adjacent endpoints.

## Abstract

A graph $G$ is called an $L_1$-graph if $d(u)+d(v)\ge|N(u)\cup N(v)\cup N(w)|-1$ for every triple of vertices $u,v,w$ where $u$ and $v$ are at distance 2 and $w\in N(u)\cap N(v)$. Asratian et al. (1996) proved that all finite connected $L_1$-graphs on at least three vertices such that $|N(u)\cap N(v)|\ge2$ for each pair of vertices $u,v$ at distance 2 are Hamiltonian, except for a simple family $\mathcal{K}$ of exceptions. We show that not all such graphs are pancyclic, but that any non-Hamiltonian cycle in such a graph can be extended to a larger cycle containing all vertices of the original cycle and at most two other vertices. We also prove a similar result for paths whose endpoints do not have any common neighbors.