$L(n)$ graphs are vertex-pancyclic and Hamilton-connected

TL;DR

This paper proves that the line graph of a specific graph construction, called $L(n)$, is both vertex-pancyclic and Hamilton-connected for all $n geq 6$, expanding understanding of cycle properties in these graphs.

Contribution

The paper establishes that $L(n)$ graphs are vertex-pancyclic and Hamilton-connected for all $n geq 6$, providing new insights into their cycle and connectivity properties.

Findings

$L(n)$ graphs are vertex-pancyclic for $n geq 6$.

$L(n)$ graphs are Hamilton-connected for $n geq 6$.

Abstract

A graph $G$ of order $n>2$ is pancyclic if $G$ contains a cycle of length $l$ for each integer $l$ with $3 \leq l \leq n $ and it is called vertex-pancyclic if every vertex is contained in a cycle of length $l$ for every $3 \leq l \leq n $. A graph $G$ of order $n > 2$ is Hamilton-connected if for any pair of distinct vertices $u$ and $v$, there is a Hamilton $u$-$v$ path, namely, there is a $u$-$v$ path of length $n-1$. The graph $ B(n)$ is a graph with the vertex set $V=\{v \ | \ v \subset [n] , | v | \in \{ 1,2 \} \} $ and the edge set $ E= \{ \{ v , w \} \ | \ v , w \in V , v \subset w $ or $ w \subset v \}$, where $[n]=\{1,2,...,n\}$. We denote by $L(n)$ the line graph of $B(n)$, that is, $L(n)=L(B(n))$. In this paper, we show that the graph $L(n)$ is vertex-pancyclic and Hamilton-connected whenever $n\geq 6$.