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### (r)-Pancyclic, (r)-Bipancyclic and Oddly (r)-Bipancyclic Graphs [article]

Abdollah Khodkar, Oliver Sawin, Lisa Mueller, WonHyuk Choi
2015 arXiv   pre-print
In this paper, using computer search, we classify all (r)-pancyclic and (r)-bipancyclic graphs with v vertices and at most v+5 edges.  ...  A graph with v vertices is (r)-pancyclic if it contains precisely r cycles of every length from 3 to v.  ...  Acknowledgement: The authors would like to thank Alexander Clifton for his suggestion to study oddly (r)-bipancyclic graphs.  ...

### Pancyclic zero divisor graph over the ring Z_n[i] [article]

Ravindra Kumar, Om Prakash
2018 arXiv   pre-print
Let Γ(Z_n[i]) be the zero divisor graph over the ring Z_n[i]. In this article, we study pancyclic properties of Γ(Z_n[i]) and Γ(Z_n[i]) for different n.  ...  Also, we prove some results in which L(Γ(Z_n[i])) and L(Γ(Z_n[i])) to be pancyclic for different values of n.  ...  If a graph contains every cycle of even length n, where 4 ď n ď p, then the graph is said to be bipancyclic. 2. When is ΓpZ n q and ΓpZ n risq is Pancyclic.  ...

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

D. Amar
1993 Discrete Mathematics
to ensure the existence of a hamiltonian cycle also imply bipancyclism, and obtain a new proof (and easier) of a theorem of Mitchem and Schmeichel (1982) .  ...  ., Applying a condition for a hamiltonian bipartite graph to be bipancyclic, Discrete Mathematics 111 (1993) 19-25.  ...  Introduction In general graphs, the following theorem of Hakimi and Schmeichel [S] gives a sufficient condition for a hamiltonian graph to be pancyclic. Theorem 1.1 (Hakimi and Schmeichel [S]).  ...

### 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 [122] .  ...  A graph is of pancyclicity if it is pancyclic.  ...  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,  ...

### Edge-fault-tolerant edge-bipancyclicity of balanced hypercubes [article]

Pingshan Li, Min Xu
2017 arXiv   pre-print
The balanced hypercube, BH_n, is a variant of hypercube Q_n. R.X.  ...  Our result is optimal with respect to the maximum number of tolerated edge faults.  ...  This research is supported by the National Natural Science Foundation of China (11571044, 61373021, 11461004) , the Fundamental Research Funds for the Central University of China .  ...

### 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 graph G is said to be Hamiltonian if it contains a cycle of length |V (G)|. The graph G is said to be pancyclic if it contains cycles of every length from 3 to |V (G)|.  ...

### Edge-bipancyclicity of star graphs under edge-fault tolerant

Min Xu, Xiao-Dong Hu, Qiang Zhu
2006 Applied Mathematics and Computation
The star graph S n is one of the most famous interconnection networks. It has been shown by Li [T.-K. Li, Cycle embedding in star graphs with edge faults, Appl. Math.  ...  Comput. 167 (2005) 891 -900] that S n contains a cycle of length from 6 to n! when the number of fault edges in the graph does not exceed n À 3.  ...  Obviously, every edge-pancyclic graph is vertex-pancyclic. A bipartite graph G is vertex-bipancyclic if for any vertex u, there exists a cycle of every even length from 4 to jV(G)j containing u.  ...

### Highly fault-tolerant cycle embeddings of hypercubes

Ming-Chien Yang, Jimmy J.M. Tan, Lih-Hsing Hsu
2007 Journal of systems architecture
The hypercube Q n is one of the most popular networks. In this paper, we first prove that the n-dimensional hypercube is 2n À 5 conditional fault-bipancyclic.  ...  In addition, if a certain node is incident with less than two healthy links, we show that an injured hypercube contains cycles of all even lengths except hamiltonian cycles with up to 2n À 3 faulty links  ...  A hamiltonian cycle is a cycle which walks through all the vertices of G. A graph is hamiltonian if it has a hamiltonian cycle.  ...

### 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
\(G\) is said to be balanced if \(|X| = |Y|\) and \(G\) 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.  ...  The graph G is said to be Hamiltonian if it contains a cycle of length V(G) . The graph G is said to be pancyclic if it contains cycles of every length from 3 to V(G) .  ...

### Cyclability and pancyclability in bipartite graphs

Mohamed El Kadi Abderrezzak, Evelyne Flandrin, Denise Amar
2001 Discrete Mathematics
Let G be a 2-connected bipartite balanced graph of order 2n and bipartition (X; Y ). Let S be a subset of X of cardinality at least 3.  ...  Under the same assumption where n + 1 is replaced by n + 3, we also prove that the graph G is S-pancyclable.  ...  The graph G is said hamiltonian if it contains a cycle through all the vertices of V (G) and pancyclic (respectively bipancyclic) if it contains cycles of every length (respectively every even length)  ...

### Page 1291 of Mathematical Reviews Vol. , Issue 95c [page]

1995 Mathematical Reviews
{For the entire collection see MR 94i:68008.} 95c:05078 05C38 05C45 Zhang, She Min (PRC-SHDN-C; Jinan) Pancyclism and bipancyclism of Hamiltonian graphs. (English summary) J. Combin. Theory Ser.  ...  The proof technique involves the recursive construction of an interesting class of graphs and an adaptation of a technique of Dirac on Hamiltonian circuits. Koduvayur R.  ...

### Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes

Tseng-Kuei Li, Chang-Hsiung Tsai, Jimmy J.M. Tan, Lih-Hsing Hsu
2003 Information Processing Letters
A bipartite graph is bipancyclic if it contains a cycle of every even length from 4 to |V (G)| inclusive. It has been shown that Q n is bipancyclic if and only if n 2.  ...  To get this result, we also prove that there exists a path of length l joining any two different vertices x and y of Q n when h(x, y) l |V (G)| − 1 and l − h(x, y) is even where h(x, y) is the Hamming  ...  Bipancyclicity is essentially a restriction of the concept of pancyclicity to bipartite graphs whose cycles are necessarily of even length.  ...

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

D. Amar
1992 Discrete Mathematics
Zf there exist two vertices x1 and x2, the distance between which is two, on a hamiltonian cycle of a bipartite graph G of order 2n, such that: d(x,) + d(xz) 3 n + 1 then G is bipancyclic except in two  ...  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.  ...

### Edge-fault-tolerant edge-bipancyclicity of hypercubes

Jun-Ming Xu, Zheng-Zhong Du, Min Xu
2005 Information Processing Letters
In this paper, we consider the problem embedding a cycle into the hypercube Q n with existence of faulty edges and show that for any edge subset F of Q n with |F | n − 1 every edge of Q n − F lies on a  ...  cycle of every even length from 6 to 2 n inclusive provided n 4 and all edges in F are not incident with the same vertex.  ...  A graph G is k-edge-fault-tolerant Hamiltonian (edge-pancyclic) if the resulting graph by deleting any k edges from G is Hamiltonian (edge-pancyclic).  ...

### Bipanconnectivity and Bipancyclicity in k-ary n-cubes

I.A. Stewart, Yonghong Xiang
2009 IEEE Transactions on Parallel and Distributed Systems
2, and we also show that when k is odd, Q k n is m-panconnected, for m ¼ nðkÀ1Þþ2kÀ6 2 , and ðk À 1Þ-pancyclic (these bounds are optimal).  ...  2009) 'Bipanconnectivity and bipancyclicity in k-ary n-cubes.', IEEE transactions on parallel and distributed systems., 20 (1). pp. 25-33.  ...  The graph G is almost-pancyclic if it contains a cycle of every possible length between 4 and n, and bipancyclic if it contains a cycle of every possible even length between 4 and n (the definition of  ...
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