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Register a new userSmall cycles, generalized prisms and Hamiltonian cycles in the Bubble-sort graph
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The Bubble-sort graph $BS_n,,ngeqslant 2$, is a Cayley graph over the symmetric group $Sym_n$ generated by transpositions from the set ${(1 2), (2 3),ldots, (n-1 n)}$. It is a bipartite graph containing all even cycles of length $ell$, where $4leqslant ellleqslant n!$. We give an explicit combinatorial characterization of all its $4$- and $6$-cycles. Based on this characterization, we define generalized prisms in $BS_n,,ngeqslant 5$, and present a new approach to construct a Hamiltonian cycle based on these generalized prisms.
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We introduce and study a $d$-dimensional generalization of Hamiltonian cycles in graphs - the Hamiltonian $d$-cycles in $K_n^d$ (the complete simplicial $d$-complex over a vertex set of size $n$). Those are the simple $d$-cycles of a complete rank, or, equivalently, of size $1 + {{n-1} choose d}$. The discussion is restricted to the fields $F_2$ and $Q$. For $d=2$, we characterize the $n$s for which Hamiltonian $2$-cycles exist. For $d=3$ it is shown that Hamiltonian $3$-cycles exist for infinitely many $n$s. In general, it is shown that there always exist simple $d$-cycles of size ${{n-1} choose d} - O(n^{d-3})$. All the above results are constructive. Our approach naturally extends to (and in fact, involves) $d$-fillings, generalizing the notion of $T$-joins in graphs. Given a $(d-1)$-cycle $Z^{d-1} in K_n^d$, ~$F$ is its $d$-filling if $partial F = Z^{d-1}$. We call a $d$-filling Hamiltonian if it is acyclic and of a complete rank, or, equivalently, is of size ${{n-1} choose d}$. If a Hamiltonian $d$-cycle $Z$ over $F_2$ contains a $d$-simplex $sigma$, then $Zsetminus sigma$ is a a Hamiltonian $d$-filling of $partial sigma$ (a closely related fact is also true for cycles over $Q$). Thus, the two notions are closely related. Most of the above results about Hamiltonian $d$-cycles hold for Hamiltonian $d$-fillings as well.
Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we show that every $4$-connected planar triangulation on $n$ vertices has $Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{Omega(n^{1/4})}$ Hamiltonian cycles.
In this series of papers, the primary goal is to enumerate Hamiltonian cycles (HCs) on the grid cylinder graphs $P_{m+1}times C_n$, where $n$ is allowed to grow whilst $m$ is fixed. In Part~I, we studied the so-called non-contractible HCs. Here, in Part~II, we proceed further on to the contractible case. We propose two different novel characterizations of contractible HCs, from which we construct digraphs for enumerating the contractible HCs. Given the impression which the computational data for $m leq 9$ convey, we conjecture that the asymptotic domination of the contractible HCs versus the non-contractible HCs, among the total number of HCs, depends on the parity of $m$.}
Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices then $G$ contains at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar $n$-vertex triangulations $G$ that do not have too many separating 4-cycles or have minimum degree 5. We show that if $G$ has $O(n/{log}_2 n)$ separating 4-cycles then $G$ has $Omega(n^2)$ Hamiltonian cycles, and if $delta(G)ge 5$ then $G$ has $2^{Omega(n^{1/4})}$ Hamiltonian cycles. Both results improve previous work. Moreover, the proofs involve a double wheel structure, providing further evidence to the above conjecture.
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 $kge 2$, 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 Bodrov{z}a-Pantic, Kwong, Doroslovav{c}ki, and Pantic for $n = 2$.
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