Distance-constrained labellings of 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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Classification of nonorientable regular embeddings of cartesian products of graphs

Discrete Mathematics ◽

10.1016/j.disc.2017.03.016 ◽

2017 ◽

Vol 340 (8) ◽

pp. 1871-1877

Author(s):

Young Soo Kwon

Keyword(s):

Cartesian Products ◽

Cartesian Products Of Graphs

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Cartesian products of graphs as subgraphs of de Bruijn graphs of dimension at least three

Discrete Applied Mathematics ◽

10.1016/s0166-218x(97)00029-2 ◽

1997 ◽

Vol 79 (1-3) ◽

pp. 3-34 ◽

Cited By ~ 2

Author(s):

Thomas Andreae ◽

Martin Hintz ◽

Michael Nölle ◽

Gerald Schreiber ◽

Gerald W. Schuster ◽

...

Keyword(s):

Cartesian Products ◽

De Bruijn Graphs ◽

De Bruijn ◽

Cartesian Products Of Graphs

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Fast Searching on Cartesian Products of Graphs

Lecture Notes in Computer Science - Theory and Applications of Models of Computation ◽

10.1007/978-3-319-55911-7_48 ◽

2017 ◽

pp. 669-683 ◽

Cited By ~ 1

Author(s):

Yuan Xue ◽

Boting Yang

Keyword(s):

Cartesian Products ◽

Fast Searching ◽

Cartesian Products Of Graphs

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The Maximum Genus of Cartesian Products of Graphs

Canadian Journal of Mathematics ◽

10.4153/cjm-1974-096-2 ◽

1974 ◽

Vol 26 (5) ◽

pp. 1025-1035 ◽

Cited By ~ 11

Author(s):

Joseph Zaks

Keyword(s):

Connected Graph ◽

Connected Components ◽

Open Disk ◽

Maximum Genus ◽

Cartesian Products ◽

Euler’S Formula ◽

Cartesian Products Of Graphs

The maximum genus γM(G) of a connected graph G has been defined in [2] as the maximum g for which there exists an embedding h : G —> S(g), where S(g) is a compact orientable 2-manifold of genus g, such that each one of the connected components of S(g) — h(G) is homeomorphic to an open disk; such an embedding is called cellular. If G is cellularly embedded in S(g), having V vertices, E edges and F faces, then by Euler's formulaV-E + F = 2-2g.

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