Counting Hamiltonian Cycles in 2-Tiled Graphs

Abstract

In 1930, Kuratowski showed that K3,3 and K5 are the only two minor-minimal nonplanar 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.

Counting Traversing Hamiltonian Cycles in Tiled Graphs

“…The following lemma is crucial in understanding of a structure of Hamiltonian cycles in tiled graphs. It is an extension of claims 1-3 of Lemma 1 from [18] to tiled graphs. Lemma 1.…”

“…In this section, we introduce the concept of a tile as presented in [18] and extend it to a definition of a tiled graph. We define the concept of k-traversing Hamiltonian cycles in tiled graphs and prove their possible existence for certain values of k.…”

“…Recently, Ðokić et al presented two algorithms for determination of the number of (all spanning unions of cycles) 2-factors in three classes of grid graphs: the thin cylinder TnC m (n), torus TG m (n), all of which had a width of m [17]. On the other hand, Vegi Kalamar et al characterized all Hamiltonian cycles in 2-tiled graphs and efficiently counted them for a fixed family of tiles [18]. Others have recently tackled other (less related) families of graphs: Fernandes et al on the Matroid Basis Graph [19], Liu et al on planar triangulations [20] and Ferber et al on Dirac Hypergraphs [21].…”

“…In the present contribution, we build on the results of [18]. The introduced k-traversing Hamiltonian cycles in tiled graphs are a generalization of zigzagging (1-traversing) and traversing (2-traversing) Hamiltonian cycles from 2-tiled graphs.…”

The structure of the 2-factor transfer digraph common for rectangular, thick cylinder and Moebius strip grid graphs

“…Robotic and biochips technology trends actualize the problem of generating and enumeration of Hamiltonian paths in grid graphs [15,16]. The counting of Hamiltonian cycles on specific grid graphs was the subject of interest in [1]- [6], [12], [13], [17] and [18]. The transfer matrix approach has been proven to be the most suitable for this and similar problems [9,11,14].…”