Counting Hamiltonian cycles in 2-tiled graphs

TL;DR

This paper introduces a method to count Hamiltonian cycles in 2-tiled graphs by leveraging a simplified characterization of large 2-crossing-critical graphs, extending previous algorithms to a broader class.

Contribution

It provides a simplified description of large 2-crossing-critical graphs and generalizes an existing Hamiltonian cycle counting algorithm to all 2-tiled graphs.

Findings

Counted Hamiltonian cycles in large 2-crossing-critical graphs.

Extended algorithm to all 2-tiled graphs.

Enhanced understanding of Hamiltonian cycles in complex graph classes.

Abstract

In 1930, Kuratowski showed that $K_{3,3}$ and $K_5$ are the only two minor-minimal non-planar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. \v{S}ir\'{a}\v{n} and Kochol showed that there are infinitely many $k$-crossing-critical graphs for any $k\ge 2$, 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 Bodro\v{z}a-Panti\'c, Kwong, Doroslova\v{c}ki, and Panti\'c for $n = 2$.