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### 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] .  ...  Suppose that G satisfies the hypothesis but does not have a bipancyclic ordering. By the main result of  , G contains a 4-cycle. Thus G contains a subgraph with a bipancyclic ordering.  ...

### Page 3542 of Mathematical Reviews Vol. , Issue 93g [page]

1993 Mathematical Reviews
T. (4-ABER) An Ore-type sufficient condition for a bipancyclic ordering. Discrete Math. 102 (1992), no. 1, 47-49.  ...  The graph G is said to have a bipancyclic ordering if the vertices of X and Y can be labeled {x;, x2,---, Xn} and {y1,2,°--,¥n}, respectively, so that for any 2<k <n, the induced subgraph with vertex set  ...

### Forthcoming papers

1991 Discrete Mathematics
A condition for a hamiltonian bipartite graph to be bipancyclic Let G be a hamiltonian bipartite graph of order 2n and let: C = (x" y" x1, y" . . , x" yn, x1) be a hamiltonian cycle of G.  ...  Choudum, On forcibly connected graphic sequences sufficient conditions connected are  ...  Choudum, On forcibly connected graphic sequences sufficient conditions connected are Forthcoming papers James A.  ...

### 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.  ...  Our result extends a classical theorem of Mitchem and Schmeichel.  ...  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  .  ...

### On dominating pair degree conditions for hamiltonicity in balanced bipartite digraphs [article]

2020 arXiv   pre-print
We prove several new sufficient conditions for hamiltonicity and bipancyclicity in balanced bipartite digraphs, in terms of sums of degrees over dominating or dominated pairs of vertices.  ...  Consequently, for all but finitely many exceptional digraphs satifying condition (B k ) of this type, hamiltonicity implies bipancyclicity: It would be therefore very interesting to know whether condition  ...  max{1, λa} < k ≤ a 2 , and for every dominating pair {u, v} of vertices in D,d(u) ≥ 2a − k and d(v) ≥ a + k, or d(u) ≥ a + k and d(v) ≥ 2a − k ,then D is bipancyclic or else a directed cycle of length  ...

### 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.  ...  Our result extends a classical theorem of Mitchem and Schmeichel.  ...  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  .  ...

### Chorded pancyclicity in k-partite graphs [article]

Daniela Ferrero, Linda Lesniak
2018 arXiv   pre-print
.≥ n_k, the condition n_1≤p 2 is necessary and sufficient for every subgraph of the complete k-partite graph K(n_1,...  ...  Our result implies that for any integers, k≥ 3 and n≥ 1, a balanced k-partite graph of order kn with has at least (k^2-k)n^2-2n(k-1)+4 2 edges is chorded pancyclic.  ...  We thank the MAA -Tensor Program for Women and Mathematics for facilitating a research visit during this project.  ...

### Ore and Erdős type conditions for long cycles in balanced bipartite graphs

2009 Discrete Mathematics & Theoretical Computer Science
Graphs and Algorithms International audience We conjecture Ore and Erdős type criteria for a balanced bipartite graph of order 2n to contain a long cycle C(2n-2k), where 0 <= k < n/2.  ...  For k = 0, these are the classical hamiltonicity criteria of Moon and Moser. The main two results of the paper assert that our conjectures hold for k = 1 as well.  ...  We would also like to thank Artur Fortuna for an inspiring conversation.  ...

### Edge condition for hamiltonicity in balanced tripartite graphs

2009 Opuscula Mathematica
We prove an analogous result for balanced tripartite graphs: If G is a balanced tripartite graph of order 3n and size at least 3n 2 − 2n + 2, then G contains cycles of all lengths.  ...  A well-known theorem of Entringer and Schmeichel asserts that a balanced bipartite graph of order 2n obtained from the complete balanced bipartite Kn,n by removing at most n − 2 edges, is bipancyclic.  ...  Acknowledgments I am happy to thank Lech Adamus for his valuable comments and discussions regarding the subject of this note.  ...

### On the compatibility of the topologies of parallel tasks and computing systems [article]

2020 arXiv   pre-print
of interconnect with a given topology for solving tasks with the same and different types of information topologies.  ...  The introduction of appropriate indexes based on the original topological model of parallel computations and on the nontraditional description of a graph by its projections is proposed and elucidated.  ...  However, the abstraction of such a model from the task's information topology and from the interconnect system topology hampers getting a complete picture for the conditionality of the task's parallelism  ...

### Page 2463 of Mathematical Reviews Vol. , Issue 94e [page]

1994 Mathematical Reviews
); Luo, Sheng Zheng (PRC-HIT; Harbin) Sufficient conditions for Hamiltonian graphs concerning neighborhood union and minimum degree.  ...  The authors show for n sufficiently large that R(K,,Z,) <n+k(k+1)(k+2)log,k. If in addition k is an odd prime power, then R(Ky, Z;,) <n+2k—2.  ...

### Transversals and Bipancyclicity in Bipartite Graph Families

2021 Electronic Journal of Combinatorics
We also show that given a family $\mathcal G$ 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  ...  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.  ...  Acknowledgements I am grateful to Ladislav Stacho for making a suggestion to consider problems of pancyclicity in the setting of graph transversals and for carefully reading this paper.  ...

### Advances on the Hamiltonian Problem - A Survey

Ronald J. Gould
2003 Graphs and Combinatorics
This article is intended as a survey, updating earlier surveys in the area.  ...  For completeness of the presentation of both particular questions and the general area, it also contains material on closely related topics such as traceable, pancyclic and hamiltonian-connected graphs  ...  Stacho  gave a sufficient condition of degree sum type for a graph to be hamiltonian. This condition generalizes several old results.  ...

### Monochromatic paths and cycles in 2-edge-colored graphs with large minimum degree [article]

József Balogh, Alexandr Kostochka, Mikhail Lavrov, Xujun Liu
2021 arXiv   pre-print
We prove that for every sufficiently large n, if n = 3t+r where r ∈{0,1,2} and G is an n-vertex graph with δ(G) ≥ (3n-1)/4, then for every 2-edge-coloring of G, either there are cycles of every length  ...  Moreover, it implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for n=3t+r where r ∈{0,1,2} and every n-vertex graph G with δ(G) ≥ 3n/4, in each 2-edge-coloring  ...  We thank the referees for their valuable comments.  ...

### Cycles in graphs and related problems

Antoni Marczyk
2008 Dissertationes Mathematicae
In  the same authors studied the Dirac-type condition for bipancyclicity. Theorem 4.63. Let G be a balanced bipartite graph of order 2n. If δ(G) ≥ (n + 1)/2, then G is bipancyclic.  ...   introduced a new Ore-type condition. Theorem 3.45.  ...  From Theorem 6.28, for any m, 2 ≤ m ≤ 3p/4 + 6, there exists a (0, l)-path of length m.  ...
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