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The hamiltonicity of bipartite graphs involving neighborhood unions

Guantao Chen, Akira Saito, Bing Wei, Xuerong Zhang
<span title="">2002</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Let G = (X; Y ) be a 2-connected balanced bipartite graph with |X | = |Y | = n.  ...  In this paper, we prove that if |N (x1) ∪ N (x2)| + |N (y1) ∪ N (y2)| ¿ n + 2 for any {x1; x2} ⊆ X and {y1; y2} ⊆ Y , then G is hamiltonian except when G is a special graph on 8 or on 12 vertices.  ...  However, according to our knowledge almost nothing has been done for using neighborhood union conditions to investigate the hamiltonicity in bipartite graphs.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(01)00227-8">doi:10.1016/s0012-365x(01)00227-8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jo7cdlrhsfflpmforsgkgzrmxu">fatcat:jo7cdlrhsfflpmforsgkgzrmxu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170922172226/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/040/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS9zMDAxMjM2NXgwMTAwMjI3OA%3D%3D.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/29/a7/29a748749a0e33bb6b4ef8c00b6629fccdcbc33e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0012-365x(01)00227-8"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Page 2018 of Mathematical Reviews Vol. , Issue 96d [page]

<span title="">1996</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The proof is not straightforward and involves the clever use of a construction called the “cycle-tree” of a graph.  ...  If G is a bipartite graph with bipartition sets of equal size, and there is a Hamilton path from any vertex in one bipartition set to any vertex in the other, then G is said to be Hamilton-laceable.  ... 
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Page 4752 of Mathematical Reviews Vol. , Issue 92i [page]

<span title="">1992</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
{For the entire collection see MR 92f:05006.} 92i:05141 05C45 Jackson, Bill (4-LNDG) Neighborhood unions and Hamilton cycles. J. Graph Theory 15 (1991), no. 4, 443-451.  ...  Summary: “For sets of vertices, we consider a form of generalized degree based on neighborhood unions.  ... 
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Page 830 of Mathematical Reviews Vol. , Issue 2003B [page]

<span title="">2003</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Hamiltonicity of bipartite graphs involving neighborhood unions.  ...  Math. 1 (1963), 163-165; MR 28 #4540] from a degree condition to a neighborhood union condition.  ... 
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Neighborhood conditions for balanced independent sets in bipartite graphs

Denise Amar, Stephan Brandt, Daniel Brito, Oscar Ordaz
<span title="">1998</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Let G be a balanced bipartite graph of order 2n and minimum degree 6(G)>~3.  ...  If the neighborhood union condition is replaced by IN(S)I >n + 2 then G is hamiltonian.  ...  Acknowledgements We thank the anonymous referee for the excellent report, which helped to improve the presentation and avoid an inaccuracy.  ... 
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Author index

<span title="">2002</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Yu, A note on fragile graphs (1-3) 41-43 Chen, G., A. Saito, B. Wei and X. Zhang, The hamiltonicity of bipartite graphs involving neighborhood unions (1-3) 45-56 Corteel, S., see A.  ...  Ho, On the half-half case of the Zarankiewicz problem (1-3) 95-104 Gy + ori, E. and R.H. Schelp, Two-edge colorings of graphs with bounded degree in both , C.W. and D.B.  ... 
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On the Ramsey number of trees versus graphs with large clique number

Ronald J. Gould, Michael S. Jacobson
<span title="">1983</span> <i title="Wiley"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ukzsjb6a6zhyxnjl6nb2mmjc6m" style="color: black;">Journal of Graph Theory</a> </i> &nbsp;
., "Extremal problems involving neighborhood unions", Journal of Graph Theory, Vol. 11, No. 4 (1987), 555-564. 24.  ...  ., "On neighborhood unions and the cycle cover number of a graph", Journal of Graph Theory, 18 (1994), No. 7, 663-672. 1995 68.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1002/jgt.3190070109">doi:10.1002/jgt.3190070109</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/za52simkf5hejaztfcxladh72e">fatcat:za52simkf5hejaztfcxladh72e</a> </span>
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Page 4714 of Mathematical Reviews Vol. , Issue 97H [page]

<span title="">1997</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
(PRC-ASBJ-S; Beijing) A new generalization of Ore’s theorem involving neighborhood unions. (English summary) Systems Sci. Math. Sci. 9 (1996), no. 2, 182-192.  ...  the Hamiltonicity exponent of directed cacti.  ... 
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The Prism of the Acyclic Orientation Graph is Hamiltonian

Gara Pruesse, Frank Ruskey
<span title="1995-03-13">1995</span> <i title="The Electronic Journal of Combinatorics"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/v5dyak6ulffqfara7hmuchh24a" style="color: black;">Electronic Journal of Combinatorics</a> </i> &nbsp;
Define a graph ${AO}(G)$ whose vertices are the acyclic orientations of $G$ and whose edges join orientations that differ by reversing the direction of a single edge.  ...  The graph ${AO}(G)$ is again connected but we give examples showing that the prism is not necessarily Hamiltonian.  ...  If the prism of a bipartite graph is Hamiltonian, then the square of the graph is Hamiltonian.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.37236/1199">doi:10.37236/1199</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/zum26xlh3rc7dlqtw63dsxccte">fatcat:zum26xlh3rc7dlqtw63dsxccte</a> </span>
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Page 1302 of Mathematical Reviews Vol. , Issue 91C [page]

<span title="">1991</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
A 1-factorization is perfect if the union of any two of its 1- factors is a Hamilton cycle.  ...  in G has at least s vertices in the union of their neighborhoods).  ... 
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Page 6195 of Mathematical Reviews Vol. , Issue 94k [page]

<span title="">1994</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
Summary: “The set of Hamilton graphs (having no loops) with |E(G)| —|V(G)| =k is denoted by I, and the number of dis- tinct cycles of a graph G is denoted by f(G). Let M(k) = max{f(G); GeET;,}.  ...  Let B,;, denote the bipartite graph with r +s vertices in one part and ¢ vertices in the other part, such that the set with r+ ¢ vertices induces a complete bipartite graph and the set with s +f vertices  ... 
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Page 7187 of Mathematical Reviews Vol. , Issue 95m [page]

<span title="">1995</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
[Liu, Xin] (PRC-ASBJ-S; Beijing); Zhu, Yong Jin (PRC-ASBJ-S; Beijing) ; Zhang, Lian Zhu (PRC-ZZTC; Zhangzhou) Distance, neighborhood unions and Hamiltonian properties in graphs.  ...  Let K, and K,,, be the complete graph with n vertices and the complete bipartite graph respectively.  ... 
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Evolutionary Hamiltonian Graph Theory [article]

Zh. G. Nikoghosyan
<span title="2012-12-17">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
LCT joins together a number of NP-complete cycle problems in graph theory.  ...  NP-completeness is the kay factor insuring (by conjecture of Cook) the generation of endless developments and great diversity around large cycles problems.  ...  Denote by d(x) the degree of a vertex x in the graph G. The neighborhood of x will be denoted by N (x).  ... 
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Graph factors and factorization: 1985–2003: A survey

Michael D. Plummer
<span title="">2007</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
In the most general sense, a factor of a graph G is just a spanning subgraph of G and a graph factorization of G is a partition of the edges of G into factors.  ...  However, as we shall see in the present paper, even this extremely general definition does not capture all the factor and factorization problems that have been studied in graph theory.  ...  Acknowledgment The author wishes to thank Hajo Broersma, Mark Ellingham, Pavol Hell, Anthony Hilton, Mikio Kano, Haruhide Matsuda and Tsuyoshi Nishimura for their kind assistance in compiling some of the  ... 
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Extremal problems for forbidden pairs that imply hamiltonicity

Ralph J. Faudree, A. Gyarfas
<span title="">1999</span> <i title="Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Gora"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g64zvtslcjdqncvrfbmvf422qi" style="color: black;">Discussiones Mathematicae Graph Theory</a> </i> &nbsp;
Let C denote the claw 1<1, 3 , N the net (a graph obtained from a J<3 by attaching a disjoint edge to each vertex of the 1< 3 ), W the wounded (a graph obtained from a /{ 3 by attaching an edge to one  ...  These extremal numbers give one measure of the relative strengths of the forbidden subgraph conditions that imply a graph is hamiltonian.  ...  Thus, G must be a a vertex disjoint union of complete bipartite graphs to avoid an induced P 4 . This completes the proof of Theorem 5.  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170808115157/http://www.renyi.hu/~gyarfas/Cikkek/95_FaudreeGyarfas_ExtremalProblemsForForbiddenPairsThatImplyHamiltonicity.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/2e/b8/2eb89d2fa76db71b60bf95d30b884942e5b3d7ea.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.7151/dmgt.1082"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>
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