Counting Hamiltonian Cycles in 2-Tiled Graphs

Alen Vegi Kalamar, Tadej Žerak and Drago Bokal
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Alen Vegi Kalamar: Department of Mathematics and Computer Science, University of Maribor, 2000 Maribor, Slovenia
Tadej Žerak: Department of Mathematics and Computer Science, University of Maribor, 2000 Maribor, Slovenia
Drago Bokal: Department of Mathematics and Computer Science, University of Maribor, 2000 Maribor, Slovenia

Mathematics, 2021, vol. 9, issue 6, 1-27

Abstract: In 1930, Kuratowski showed that K 3 , 3 and K 5 are the only two minor-minimal nonplanar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. Širá? and Kochol showed that there are infinitely many k -crossing-critical graphs for any k ? 2 , even if restricted to simple 3-connected graphs. Recently, 2-crossing-critical graphs have been completely characterized by Bokal, Oporowski, Richter, and Salazar. We present a simplified description of large 2-crossing-critical graphs and use this simplification to count Hamiltonian cycles in such graphs. We generalize this approach to an algorithm counting Hamiltonian cycles in all 2-tiled graphs, thus extending the results of Bodroža-Panti?, Kwong, Doroslova?ki, and Panti?.

Keywords: crossing number; crossing-critical graph; Hamiltonian cycle (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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