A graph of order n ≥ 3 is said to be pancyclic if it contains a cycle of each length from 3 to n. A chord of a cycle is an edge between two nonadjacent vertices of the cycle. A chorded cycle is a cycle containing at least one chord. We define a graph of order n ≥ 4 to be chorded pancyclic if it contains a chorded cycle of each length from 4 to n. In this article, we prove the following: If G is...

A digraph obtained by replacing each edge of a complete multipartite graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a complete multipartite digraph. Such a digraph D is called ordinary if for any pair X, Y of its partite sets the set of arcs with end vertices in X ∪ Y coincides with X × Y = {x, y) : x ∈ X, y ∈ Y } or Y ×X or X×Y ∪Y ×X. We characterize a...

Alspach, B. and D. Hare, Edge-pancyclic block-intersection graphs, Discrete Mathematics 97 (1991) 17-24. It is shown that the block-intersection graph of both a balanced incomplete block design with block size at least 3 and A = 1, and a transversal design is edge-pancyclic.

An arc in a tournament T with n 3 vertices is called k-pancyclic, if it belongs to a cycle of length for all k n. In this paper, we show that each s-strong tournament with s 3 contains at least s + 1 vertices whose out-arcs are 4-pancyclic. © 2006 Elsevier B.V. All rights reserved.

A graph G is vertex pancyclic if for each vertex v ∈ V (G), and for each integer k with 3 ≤ k ≤ |V (G)|, G has a k-cycle Ck such that v ∈ V (Ck). Let s ≥ 0 be an integer. If the removal of at most s vertices in G results in a vertex pancyclic graph, we say G is an s-vertex pancyclic graph. Let G be a simple connected graph that is not a path, cycle or K1,3. Let l(G) = max{m : G has a divalent p...

A graph G on n vertices is Hamiltonian if it contains a cycle of length n and pancyclic if it contains cycles of length ` for all 3 ≤ ` ≤ n. Write α(G) for the independence number of G, i.e. the size of the largest subset of the vertex set that does not contain an edge, and κ(G) for the (vertex) connectivity, i.e. the size of the smallest subset of the vertex set that can be deleted to obtain a...

A digraph D=(V(D),A(D)) of order n≥3 is pancyclic, whenever D contains a directed cycle length k for each k∈{3,…,n}; and vertex-pancyclic iff, vertex v∈V(D) k∈{3,…,n}, passing through v. Let D1, D2, …, Dk be collection pairwise disjoint digraphs. The generalized sum (g.s.) Dk, denoted by ⊕i=1kDi or D1⊕D2⊕⋯⊕Dk, the set all digraphs satisfying: (i) V(D)=⋃i=1kV(Di), (ii) D〈V(Di)〉≅Di i=1,2,…,k, (ii...

Let $T$ be a non-trivial tournament. An arc is emph{$t$-pancyclic} in $T$, if it is contained in a cycle of length $ell$ for every $tleq ell leq |V(T)|$. Let $p^t(T)$ denote the number of $t$-pancyclic arcs in $T$ and $h^t(T)$ the maximum number of $t$-pancyclic arcs contained in the same Hamiltonian cycle of $T$. Moon ({em J. Combin. Inform. System Sci.}, {bf 19} (1994), 207-214) showed that $...

A graph on n vertices is called pancyclic if it contains a cycle of length ` for all 3 ≤ ` ≤ n. In 1972, Erdős proved that if G is a Hamiltonian graph on n > 4k vertices with independence number k, then G is pancyclic. He then suggested that n = Ω(k) should already be enough to guarantee pancyclicity. Improving on his and some other later results, we prove that there exists a constant c such th...