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### A condition for a hamiltonian bipartite graph to be bipancyclic

D. Amar
1992 Discrete Mathematics
But if the graph is supposed to be hamiltonian, we need weaker conditions in order for the graph to be pancyclic [l, 31 . When G is a balanced bipartite graph, we have analogous results [9-111.  ...  Introduction Let G be a finite graph of order n. Various sufficient conditions for a graph to be hamiltonian have been given in terms of vertex-degree and size of the graph [5-81.  ...

### Applying a condition for a hamiltonian bipartite graph to be bipancyclic

D. Amar
1993 Discrete Mathematics
., Applying a condition for a hamiltonian bipartite graph to be bipancyclic, Discrete Mathematics 111 (1993) 19-25.  ...  sufficient for the graph to be bipancyclic.  ...  Sufficient conditions for a bipartite graph to be bipancyclic In bipartite graphs, the following theorem gives a condition Theorem 1.1 for a hamiltonian bipartite graph to be bipancyclic. similar to that  ...

### Spectral Conditions for the Bipancyclic Bipartite Graphs [article]

Rao Li
2021 arXiv   pre-print
G is said to be bipancyclic if it contains cycles of every even length from 4 to |V(G)|. In this note, we present spectral conditions for the bipancyclic bipartite graphs.  ...  Let G = (X, Y; E) be a bipartite graph with two vertex partition subsets X and Y. G is said to be balanced if |X| = |Y|.  ...  The bipartite graph G is said to be balanced if |V 1 | = |V 2 |. Clearly, if a bipartite graph is Hamiltonian, then it must be balanced.  ...

### Spectral conditions for the Bipancyclic Bipartite graphs

Rao Li, Department of mathematical sciences, University of South Carolina Aiken, Aiken, SC 29801, USA
2021 Open Journal of Discrete Applied Mathematics
In this note, we present spectral conditions for the bipancyclic bipartite graphs.  ...  Let \(G = (X, Y; E)\) be a bipartite graph with two vertex partition subsets \(X\) and \(Y\).  ...  The bipartite graph G is said to be balanced if V 1 = V 2 . Clearly, if a bipartite graph is Hamiltonian, then it must be balanced.  ...

### An Ore-type sufficient condition for a bipancyclic ordering

George R.T. Hendry
1992 Discrete Mathematics
., An Ore-type sufficient condition for a bipancyclic ordering, Discrete Mathematics 102 (1992) 47-49.  ...  A weaker sufficient condition for hamiltonicity was given by Chvatal [l] .  ...  Let G(X, Y, E) be a balanced bipartite graph of order 2n 2 4 satisfying a(G)>n+l. (1) 0012-365X/92/\$05.00 0 1992-Elsevier Science Publishers B.V.  ...

### On the cycle structure of hamiltonian k-regular bipartite graphs of order 4k [article]

2009 arXiv   pre-print
Moreover, if such a cycle can be chosen to omit a pair of adjacent vertices, then G is bipancyclic.  ...  It is shown that a hamiltonian n/2-regular bipartite graph G of order 2n>8 contains a cycle of length 2n-2.  ...  In  , Entringer and Schmeichel gave a sufficient condition for a hamiltonian bipartite graph to be bipancyclic. Theorem 1.  ...

### Highly fault-tolerant cycle embeddings of hypercubes

Ming-Chien Yang, Jimmy J.M. Tan, Lih-Hsing Hsu
2007 Journal of systems architecture
In this paper, we first prove that the n-dimensional hypercube is 2n À 5 conditional fault-bipancyclic.  ...  That is, an injured hypercube with up to 2n À 5 faulty links has a cycle of length l for every even 4 6 l 6 2 n when each node of the hypercube is incident with at least two healthy links.  ...  A bipartite graph G is k conditional edge fault-tolerant bipancyclic (abbreviated as k conditional fault-bipancyclic) if G À F is bipancyclic for every F with jFj 6 k under the condition that every vertex  ...

### Transversals and bipancyclicity in bipartite graph families [article]

2021 arXiv   pre-print
A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number of vertices in the graph.  ...  We also show that given a family 𝒢 of n bipartite graphs on a common set X of 2n vertices meeting the same degree conditions, there exists a perfect matching on X that uses exactly one edge from each  ...  Therefore, Theorem 1.2 may be summarized by stating that a balanced bipartite graph meeting certain minimum degree conditions is bipancyclic.  ...

### A Meyniel-Type Condition for Bipancyclicity in Balanced Bipartite Digraphs

2018 Graphs and Combinatorics
We prove that a strongly connected balanced bipartite digraph D of order 2a, a≥3, satisfying d(u)+d(v)≥ 3a for every pair of vertices u,v with a common in-neighbour or a common out-neighbour, is either  ...  bipancyclic or a directed cycle of length 2a.  ...  Acknowledgments The author is grateful to an anonymous referee for spotting a critical mistake in an earlier version of the manuscript.  ...

### Survey on path and cycle embedding in some networks

Jun-Ming Xu, Meijie Ma
2009 Frontiers of Mathematics in China
Obviously, an edgepancyclic graph is certainly vertex-pancyclic. A graph G is said to be hamiltonian connected if there exists a hamiltonian path between any two vertices of G  .  ...  A graph G of order n is said to be panconnected if for any pair of different vertices x and y with distance d in G, there exist xy-paths of every length from d to n − 1 [5, 151] . A graph  ...  Acknowledgements As an invited speaker, the first author partially reported this work at the International Workshop on Structures and Cycles in Graphs, 5-9 July 2005, Wuhan,  ...

### New sufficient conditions for bipancyclic bipartite graphs

Mohamed El Kadi Abderrezzak, Evelyne Flandrin
2004 Discrete Mathematics
We give here two su cient conditions for a bipartite balanced graph of order 2n to be bipancyclic.  ...  one is for bipartite balanced traceable graphs containing an hamiltonian path whose extremities are nonadjacent and have degree sum at least n + 1.  ...  We then give two new su cient conditions for a bipartite balanced graph to be bipancyclic.  ...

### Bipancyclic subgraphs in random bipartite graphs [article]

Yilun Shang
2012 arXiv   pre-print
A bipartite graph on 2n vertices is bipancyclic if it contains cycles of all even lengths from 4 to 2n.  ...  In this paper we prove that the random bipartite graph G(n,n,p) with p(n)≫ n^-2/3 asymptotically almost surely has the following resilience property: Every Hamiltonian subgraph G' of G(n,n,p) with more  ...  A variety of sufficient conditions for a Hamiltonian bipartite graph to be bipancyclic have been studied in the literature, including [1, 2, 12, 13] and  .  ...

### Bipancyclic Subgraphs in Random Bipartite Graphs

Yilun Shang
2013 International Journal of Open Problems in Computer Science and Mathematics
A bipartite graph on 2n vertices is bipancyclic if it contains cycles of all even lengths from 4 to 2n.  ...  In this paper we prove that the random bipartite graph G(n, n, p) with p(n) n −2/3 asymptotically almost surely has the following resilience property: Every Hamiltonian subgraph G of G(n, n, p) with more  ...  A variety of sufficient conditions for a Hamiltonian bipartite graph to be bipancyclic have been studied in the literature, including [1, 2, 12, 13] and  .  ...

### Page 5360 of Mathematical Reviews Vol. , Issue 92j [page]

1992 Mathematical Reviews
A bipartite graph of even order is bipancyclic if it contains cycles of all possible even lengths. The authors consider here a conjecture due to J. Mitchem and E. F.  ...  The authors generalize this result by proving it to be true when mp < n < d(u)+ d(v)+a, where a > 4, and they obtain bounds for c(G) and n— c(G) for such graphs.  ...

### Linear array and ring embeddings in conditional faulty hypercubes

Chang-Hsiung Tsai
2004 Theoretical Computer Science
Based on this requirement, we prove that Qn contains a hamiltonian path joining any two di erent colored vertices even if it has up to 2n − 5 edge faults.  ...  The n-dimensional hypercube Qn is a graph having 2 n vertices labeled from 0 to 2 n − 1. Two vertices are connected by an edge if their binary labels di er in exactly one bit position.  ...  A bipartite graph G = (V 0 ; V 1 ; E) is said to be equitable if and only if |V 0 | = |V 1 |, and Simmons  introduced the concept of hamiltonian laceability for those equitable bipartite graphs.  ...
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