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(r)-Pancyclic, (r)-Bipancyclic and Oddly (r)-Bipancyclic Graphs
[article]
2015
arXiv
pre-print
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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. ...
arXiv:1510.03052v1
fatcat:nrtngmaq2vbifp2oatnnt7bnuq
Pancyclic zero divisor graph over the ring Z_n[i]
[article]
2018
arXiv
pre-print
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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. ...
arXiv:1806.08261v1
fatcat:hwn6mwf5b5fs7dpr7ni3ntorsi
Applying a condition for a hamiltonian bipartite graph to be bipancyclic
1993
Discrete Mathematics
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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]). ...
doi:10.1016/0012-365x(93)90136-h
fatcat:olgtucxxjzeabeuhdclxpfjufm
Survey on path and cycle embedding in some networks
2009
Frontiers of Mathematics in China
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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, ...
doi:10.1007/s11464-009-0017-5
fatcat:sizti6s4pvfazgyrijnfvomwra
Edge-fault-tolerant edge-bipancyclicity of balanced hypercubes
[article]
2017
arXiv
pre-print
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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 . ...
arXiv:1606.05152v4
fatcat:3ld4b3nrfrbf7bwvfkslfnq5mm
Spectral Conditions for the Bipancyclic Bipartite Graphs
[article]
2021
arXiv
pre-print
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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)|. ...
arXiv:2106.13741v1
fatcat:yzwp66subjgellbtc5yfdp6yrm
Edge-bipancyclicity of star graphs under edge-fault tolerant
2006
Applied Mathematics and Computation
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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. ...
doi:10.1016/j.amc.2006.05.117
fatcat:3p27p3ucfzgvnfkzhzlfcs47he
Highly fault-tolerant cycle embeddings of hypercubes
2007
Journal of systems architecture
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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. ...
doi:10.1016/j.sysarc.2006.10.008
fatcat:wzjf4of2wvcq7gm6kgy3ewzjyq
Spectral conditions for the Bipancyclic Bipartite graphs
2021
Open Journal of Discrete Applied Mathematics
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\(G\) 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) . ...
doi:10.30538/psrp-odam2021.0058
fatcat:46qi5a747vettjh74guuvrxs7i
Cyclability and pancyclability in bipartite graphs
2001
Discrete Mathematics
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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) ...
doi:10.1016/s0012-365x(00)00425-8
fatcat:43eqgxxyfba4fhamxs37grxho4
Page 1291 of Mathematical Reviews Vol. , Issue 95c
[page]
1995
Mathematical Reviews
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{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
2003
Information Processing Letters
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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. ...
doi:10.1016/s0020-0190(03)00258-8
fatcat:enyb6spzmzg3jcqk3g7i4bctla
A condition for a hamiltonian bipartite graph to be bipancyclic
1992
Discrete Mathematics
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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. ...
doi:10.1016/0012-365x(92)90116-w
fatcat:2ukehq4yyzex5iiiflo7ow4pli
Edge-fault-tolerant edge-bipancyclicity of hypercubes
2005
Information Processing Letters
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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). ...
doi:10.1016/j.ipl.2005.06.006
fatcat:i4w2xmywezek5d3hkqgl7zjvsm
Bipanconnectivity and Bipancyclicity in k-ary n-cubes
2009
IEEE Transactions on Parallel and Distributed Systems
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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 ...
doi:10.1109/tpds.2008.45
fatcat:d6ftmd4ppvba3ltfwqiu4hctfi
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