On Cartesian Products of Cyclic Orthogonal Double Covers of Circulants

LetHbe a graph onnvertices and𝒢a collection ofnsubgraphs ofH, one for each vertex, where𝒢is an orthogonal double cover (ODC) ofHif every edge ofHoccurs in exactly two members of𝒢and any two members share an edge whenever the corresponding vertices are adjacent inHand share no edges whenever the corresponding vertices are nonadjacent inH. In this paper, we are concerned with the Cartesian product of symmetric starter vectors of orthogonal double covers of the complete bipartite graphs and using this method to construct ODCs by new disjoint unions of complete bipartite graphs.

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On Cartesian Products of Any Finite Number of Orthogonal Double Covers

British Journal of Applied Science & Technology ◽

10.9734/bjast/2014/8689 ◽

2014 ◽

Vol 4 (15) ◽

pp. 2231-2240

Author(s):

R. El-Shanawany

Keyword(s):

Finite Number ◽

Cartesian Products ◽

Double Covers

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Generalization of the Symmetric Starter of the Orthogonal Double Covers of the Complete Bipartite Graph

Menoufia Journal of Electronic Engineering Research ◽

10.21608/mjeer.2006.64856 ◽

2006 ◽

Vol 16 (2) ◽

pp. 171-177

Author(s):

S. A. El-Serafi ◽

R. A. El-Shanawany ◽

M. Sh. Higazy

Keyword(s):

Bipartite Graph ◽

Complete Bipartite Graph ◽

Double Covers

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Double cover K3 surfaces of Hirzebruch surfaces

Advances in Geometry ◽

10.1515/advgeom-2020-0034 ◽

2021 ◽

Vol 21 (2) ◽

pp. 221-225

Author(s):

Taro Hayashi

Keyword(s):

Rational Surface ◽

K3 Surfaces ◽

Double Cover ◽

Double Covering ◽

Smooth Surfaces ◽

Branch Locus ◽

Hirzebruch Surfaces ◽

Double Covers

Abstract General K3 surfaces obtained as double covers of the n-th Hirzebruch surfaces with n = 0, 1, 4 are not double covers of other smooth surfaces. We give a criterion for such a K3 surface to be a double covering of another smooth rational surface based on the branch locus of double covers and fibre spaces of Hirzebruch surfaces.

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Peg Solitaire on Cartesian Products of Graphs

Graphs and Combinatorics ◽

10.1007/s00373-021-02289-7 ◽

2021 ◽

Vol 37 (3) ◽

pp. 907-917

Author(s):

Martin Kreh ◽

Jan-Hendrik de Wiljes

Keyword(s):

Cartesian Product ◽

Cartesian Products ◽

Connected Graphs ◽

Grid Graphs ◽

Cartesian Products Of Graphs

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

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