The 2‐hamiltonian cubes of graphs

Koh, Khee Meng; Teo, Kee L.

doi:10.1002/jgt.3190130609

Cited by 5 publications

(2 citation statements)

References 10 publications

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“…Whether the cube is 1-hamiltonian-connected, i.e., it still remains hamiltonian-connected after the removal of any one vertex, was characterized for trees by Lesniak [21] and for connected graphs by Schaar [28]. Characterizations of connected graphs whose cubes are p-hamiltonian for p ≤ 3 were also made in [20,28], and strong hamiltonian properties of the cube of a 2-edge connected graph were studied in [23].…”

Section: Strong Hamiltonian Propertiesmentioning

confidence: 99%

Disjoint path covers in cubes of connected graphs

Park

Ihm

2014

Discrete Mathematics

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Given a graph G, and two vertex sets S and T of size k each, a many-tomany k-disjoint path cover of G joining S and T is a collection of k disjoint paths between S and T that cover every vertex of G. It is classified as paired if each vertex of S must be joined to a designated vertex of T , or unpaired if there is no such constraint. In this article, we first present a necessary and sufficient condition for the cube of a connected graph to have a paired 2-disjoint path cover. Then, a corresponding condition for the unpaired type of 2-disjoint path cover problem is immediately derived. It is also shown that these results can easily be extended to determine if the cube of a connected graph has a hamiltonian path from a given vertex to another vertex that passes through a prescribed edge.

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Section: Strong Hamiltonian Propertiesmentioning

confidence: 99%

Disjoint path covers in cubes of connected graphs

Park

Ihm

2014

Discrete Mathematics

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“…It was also shown that the cube of a connected graph having order at least four is 1-hamiltonian [5]. Trees and connected graphs whose cubes are 1-hamiltonian-connected were characterized respectively in [18,25], while connected graphs whose cubes are p-hamiltonian for p ≤ 3 were classified in [17,25]. In [20], strong hamiltonian properties of the cube of a 2-edge-connected graph were also studied.…”

Section: Preliminariesmentioning

confidence: 99%

Single-source three-disjoint path covers in cubes of connected graphs

Park

Ihm

2013

Information Processing Letters

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A k-disjoint path cover (k-DPC for short) of a graph is a set of k internally vertex-disjoint paths from given sources to sinks that collectively cover every vertex in the graph. In this paper, we establish a necessary and sufficient condition for the cube of a connected graph to have a 3-DPC joining a single source to three sinks. We also show that the cube of a connected graph always has a 3-DPC joining arbitrary two vertices.

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Hamiltonian Submanifolds of Regular Polytopes

Effenberger

Kühnel

2009

Discrete Comput Geom

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We investigate polyhedral 2k-manifolds as subcomplexes of the boundary complex of a regular polytope. We call such a subcomplex k-Hamiltonian if it contains the full k-skeleton of the polytope. Since the case of the cube is well known and since the case of a simplex was also previously studied (these are so-called superneighborly triangulations), we focus on the case of the cross polytope and the sporadic regular 4-polytopes. By our results the existence of 1-Hamiltonian surfaces is now decided for all regular polytopes. Furthermore we investigate 2-Hamiltonian 4-manifolds in the d-dimensional cross polytope. These are the "regular cases" satisfying equality in Sparla's inequality. In particular, we present a new example with 16 vertices which is highly symmetric with an automorphism group of order 128. Topologically it is homeomorphic to a connected sum of seven copies of S 2 × S 2 . By this example all regular cases of n vertices with n < 20 or, equivalently, all cases of regular d-polytopes with d ≤ 9 are now decided.

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The 2‐hamiltonian cubes of graphs

Cited by 5 publications

References 10 publications

Disjoint path covers in cubes of connected graphs

Disjoint path covers in cubes of connected graphs

Single-source three-disjoint path covers in cubes of connected graphs

Hamiltonian Submanifolds of Regular Polytopes

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