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

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 $n\geq 6$.Comment: 7 pages. 1 figur

A result on Hamiltonian line graphs involving restrictions on induced subgraphs

It is shown that the existence of a Hamilton cycle in the line graph of a graph G can be ensured by imposing certain restrictions on certain induced subgraphs of G. Thereby a number of known results on hamiltonian line graphs are improved, including the earliest results in terms of vertex degrees. One particular consequence is that every graph of diameter 2 and order at least 4 has a hamiltonian line graph

Long cycles and paths in distance graphs

AbstractFor n∈N and D⊆N, the distance graph PnD has vertex set {0,1,…,n−1} and edge set {ij∣0≤i,j≤n−1,|j−i|∈D}. Note that the important and very well-studied circulant graphs coincide with the regular distance graphs.A fundamental result concerning circulant graphs is that for these graphs, a simple greatest common divisor condition, their connectivity, and the existence of a Hamiltonian cycle are all equivalent. Our main result suitably extends this equivalence to distance graphs. We prove that for a finite set D of order at least 2, there is a constant cD such that the greatest common divisor of the integers in D is 1 if and only if for every n, PnD has a component of order at least n−cD if and only if for every n≥cD+3, PnD has a cycle of order at least n−cD. Furthermore, we discuss some consequences and variants of this result

Pancyclicity of 4-connected {Claw, Generalized Bull}-free Graphs

A graph G is pancyclic if it contains cycles of each length ℓ, 3 ≤ ℓ ≤ |V (G)|. The generalized bull B(i, j) is obtained by associating one endpoint of each of the paths P i+1 and P j+1 with distinct vertices of a triangle. Gould, Luczak and Pfende