Counting Hamiltonian cycles in 2-tiled graphs

2021/02/16 by Alen Vegi Kalamar, Kalamar, Alen Vegi, Tadej Žerak +3
Mathematics · #05C30 #05C38 #Combinatorics (math.CO) #FOS: Mathematics #G.2.1 #G.2.2 #acm:05C30 #acm:05C38 #math.CO #msc:05C30 #msc:05C38

paper · pdf · doi:10.48550/arxiv.2102.07985

19 pages

arxiv created 2021/02/16 · arxiv updated 2021/02/18

Abstract

In 1930, Kuratowski showed that K_3,3 and K₅ are the only two minor-minimal non-planar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. Širáň 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-Pantić, Kwong, Doroslovački, and Pantić for n = 2.

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