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

2015
*
arXiv
*
pre-print

In this paper, using computer search, we classify all (r)-

arXiv:1510.03052v1
fatcat:nrtngmaq2vbifp2oatnnt7bnuq
*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*. ...##
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Pancyclic zero divisor graph over the ring Z_n[i]
[article]

2018
*
arXiv
*
pre-print

Let Γ(Z_n[i]) be the zero divisor

arXiv:1806.08261v1
fatcat:hwn6mwf5b5fs7dpr7ni3ntorsi
*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*. ...##
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Applying a condition for a hamiltonian bipartite graph to be bipancyclic

1993
*
Discrete Mathematics
*

to ensure the existence

doi:10.1016/0012-365x(93)90136-h
fatcat:olgtucxxjzeabeuhdclxpfjufm
*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]). ...##
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Survey on path and cycle embedding in some networks

2009
*
Frontiers of Mathematics in China
*

Obviously, an edgepancyclic

doi:10.1007/s11464-009-0017-5
fatcat:sizti6s4pvfazgyrijnfvomwra
*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, ...##
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Edge-fault-tolerant edge-bipancyclicity of balanced hypercubes
[article]

2017
*
arXiv
*
pre-print

The balanced hypercube, BH_n, is a variant

arXiv:1606.05152v4
fatcat:3ld4b3nrfrbf7bwvfkslfnq5mm
*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 . ...##
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Spectral Conditions for the Bipancyclic Bipartite Graphs
[article]

2021
*
arXiv
*
pre-print

G is said to be

arXiv:2106.13741v1
fatcat:yzwp66subjgellbtc5yfdp6yrm
*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)|. ...##
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Edge-bipancyclicity of star graphs under edge-fault tolerant

2006
*
Applied Mathematics and Computation
*

The star

doi:10.1016/j.amc.2006.05.117
fatcat:3p27p3ucfzgvnfkzhzlfcs47he
*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. ...##
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Highly fault-tolerant cycle embeddings of hypercubes

2007
*
Journal of systems architecture
*

The hypercube Q n is one

doi:10.1016/j.sysarc.2006.10.008
fatcat:wzjf4of2wvcq7gm6kgy3ewzjyq
*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. ...##
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Spectral conditions for the Bipancyclic Bipartite graphs

2021
*
Open Journal of Discrete Applied Mathematics
*

\(G\) is said to be balanced if \(|X| = |Y|\)

doi:10.30538/psrp-odam2021.0058
fatcat:46qi5a747vettjh74guuvrxs7i
*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) . ...##
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Cyclability and pancyclability in bipartite graphs

2001
*
Discrete Mathematics
*

Let G be a 2-connected bipartite balanced

doi:10.1016/s0012-365x(00)00425-8
fatcat:43eqgxxyfba4fhamxs37grxho4
*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) ...##
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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. ...##
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Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes

2003
*
Information Processing Letters
*

A bipartite

doi:10.1016/s0020-0190(03)00258-8
fatcat:enyb6spzmzg3jcqk3g7i4bctla
*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. ...##
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A condition for a hamiltonian bipartite graph to be bipancyclic

1992
*
Discrete Mathematics
*

Zf there exist two vertices x1

doi:10.1016/0012-365x(92)90116-w
fatcat:2ukehq4yyzex5iiiflo7ow4pli
*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. ...##
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Edge-fault-tolerant edge-bipancyclicity of hypercubes

2005
*
Information Processing Letters
*

In this paper, we consider the problem embedding a cycle into the hypercube Q n with existence

doi:10.1016/j.ipl.2005.06.006
fatcat:i4w2xmywezek5d3hkqgl7zjvsm
*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*). ...##
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Bipanconnectivity and Bipancyclicity in k-ary n-cubes

2009
*
IEEE Transactions on Parallel and Distributed Systems
*

2,

doi:10.1109/tpds.2008.45
fatcat:d6ftmd4ppvba3ltfwqiu4hctfi
*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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