Hamiltonian Laceability of Hypercubes Without Isometric Subgraphs

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F16%3A10332705" target="_blank" >RIV/00216208:11320/16:10332705 - isvavai.cz</a>
Result on the web
<a href="http://dx.doi.org/10.1007/s00373-016-1728-5" target="_blank" >http://dx.doi.org/10.1007/s00373-016-1728-5</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1007/s00373-016-1728-5" target="_blank" >10.1007/s00373-016-1728-5</a>

Result language
angličtina
Original language name
Hamiltonian Laceability of Hypercubes Without Isometric Subgraphs
Original language description
Locke conjectured that the n-dimensional hypercube Q(n) with the set F of 2k removed vertices half from each bipartition set, where n >= k + 2 and k > 1, is Hamiltonian. So far the conjecture remains open although partial results are known; some of them with additional conditions on the set F. We explore Hamiltonian properties of Q(n) - F if the set of faulty vertices F forms either an isometric cycle of Q(n) or an isometric tree of Q(n). We study a more general problem. A bipartite graph G is Hamiltonian laceable if either (a) its bipartition sets are of equal size and for each pair of vertices x, y from different bipartition sets there exists a Hamiltonian path between x and y, or (b) its bipartition sets differ in sizes by one and for each pair of vertices x, y from the larger bipartition set there exists a Hamiltonian path between x and y. In particular, we show that if C is an isometric cycle in Q(n) for n >= 5, then Q(n) - V(C) is Hamiltonian laceable. This allows us to remove up to 2n faulty vertices. Furthermore, if T is balanced isometric tree in Q(n), then for n >= 4 the graph Q(n) - V(T) is Hamiltonian laceable. Finally, if T is an almost-balanced isometric tree in Q(n), then for n >= 5 the graph Q(n) - V(T) is Hamiltonian laceable. Thus our results support Locke hypothesis.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
BA - General mathematics
OECD FORD branch
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Project
<a href="/en/project/GA14-10799S" target="_blank" >GA14-10799S: Hybercubic, graph and hypergraph structures</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Publication year
2016
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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