Solutions to Problems

Hamiltonian cycles and paths in vertex-transitive graphs with abelian and nilpotent groups

A note on the middle levels problem

In previous work, Feldmann and Mysliwietz [3] showed that every shuffle-exchange graph contains a Hamilton path. In this paper, we show that there is a linear time ranking algorithm associated with that Hamilton path. The ranking algorithm returns the rank-position of each node of the graph with respect to the order given by the Hamilton path. Ranking algorithms are important for they can be applied to yield efficient implementations of certain network emulations using SIMD-style parallel algorithms for translating node labels.KeywordsCayley GraphHamilton PathRanking AlgorithmInitial NodeGray CodeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

For a connected graph, a path containing all vertices is known as a Hamiltonian path. For general graphs, there is no known necessary and sufficient condition for the existence of Hamiltonian paths and the complexity of finding a Hamiltonian path in general graphs is NP-Complete. We present a necessary and sufficient condition for the existence of Hamiltonian paths in 2-trees. Using our characterization, we also present a linear-time algorithm for the existence of Hamiltonian paths in 2-trees. Our characterization is based on a deep understanding of the structure of 2-trees and the combinatorics presented here may be used in other combinatorial problems restricted to 2-trees.

An undirected graph $G$ is $d$-degenerate if every subgraph of $G$ has a vertex of degree at most $d$. By the classical theorem of Erdös and Gallai from 1959, every graph of degeneracy $d>1$ contains a cycle of length at least $d+1$. The proof of Erdös and Gallai is constructive and can be turned into a polynomial time algorithm constructing a cycle of length at least $d+1$. But can we decide in polynomial time whether a graph contains a cycle of length at least $d+2$? An easy reduction from Hamiltonian Cycle provides a negative answer to this question: Deciding whether a graph has a cycle of length at least $d+2$ is NP-complete. Surprisingly, the complexity of the problem changes drastically when the input graph is 2-connected. In this case we prove that deciding whether $G$ contains a cycle of length at least $d+k$ can be done in time $2^{\mathcal{O}(k)}\cdot|V(G)|^{\mathcal{O}(1)}$. In other words, deciding whether a 2-connected $n$-vertex $G$ contains a cycle of length at least $d+\log{n}$ can be done in polynomial time. Similar algorithmic results hold for long paths in graphs. We observe that deciding whether a graph has a path of length at least $d+1$ is NP-complete. However, we prove that if graph $G$ is connected, then deciding whether $G$ contains a path of length at least $d+k$ can be done in time $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$. We complement these results by showing that the choice of degeneracy as the “above guarantee parameterization” is optimal in the following sense: For any $\varepsilon>0$ it is NP-complete to decide whether a connected (2-connected) graph of degeneracy $d$ has a path (cycle) of length at least $(1+\varepsilon)d$.

An Õ(2 n) volume molecular algorithm for Hamiltonian path

A geometric graph is a graph embedded in the plane in such a way that vertices correspond to points in general position and edges correspond to segments connecting the appropriate points. A noncrossing Hamiltonian path in a geometric graph is a Hamiltonian path which does not contain any intersecting pair of edges. In the paper, we study a problem asked by Micha Perles: Determine a function h, where h(n) is the largest number k such that when we remove arbitrary set of k edges from a complete geometric graph on n vertices, the resulting graph still has a noncrossing Hamiltonian path. We prove that h(n) = Ω(√n). We also determine the function exactly in case when the removed edges form a star or a matching, and give asymptotically tight bounds in case they form a clique.

Inclusion and exclusion algorithm for the Hamiltonian path problem

A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In the past, we proved the Hamiltonian path and cycle problems for general supergrid graphs to be NP-complete. However, they are still open for solid supergrid graphs. In this paper, first we will verify the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of solid supergrid graphs. Next, we show that C-shaped supergrid graphs are Hamiltonian connected except in a few conditions. For these excluding conditions of Hamiltonian connectivity, we compute their longest paths. Then, we design a linear-time algorithm to solve the longest path problem in these graphs. The Hamiltonian connectivity of C-shaped supergrid graphs can be applied to compute the optimal stitching trace of computer embroidery machines, and construct the minimum printing trace of 3D printers with a C-like component being printed.

We consider a problem formulated by Marco Buratti concerning Hamiltonian paths in the complete graph on $Z_p$, $p$ an odd prime.

A direct algorithmic procedure is described for computing Hamiltonian circuits in graphs which satisfy a condition due to Chvatal. For a graph with p vertices this procedure works in O(p 3) operations. It is suggested how this procedure might be used heuristically to compute long paths in general graphs.

Paths and circuits in finite groups

We investigate hamiltonian properties of Pm × Cn, m ≥ 2 and even n ≥ 4, which is bipartite, in the presence of faulty vertices and/or edges. We show that Pm × Cn with n even is strongly hamiltonian-laceable if the number of faulty elements is one or less. When the number of faulty elements is two, it has a fault-free cycle of length at least mn - 2 unless both faulty elements are contained in the same partite vertex set; otherwise, it has a fault-free cycle of length mn - 4. A sufficient condition is derived for the graph with two faulty edges to have a hamiltonian cycle. By applying fault-hamiltonicity of Pm × Cn to a two-dimensional torus Cm × Cn, we obtain interesting hamiltonian properties of a faulty Cm × Cn.

A note on the Hamiltonian Circuit Problem on directed path graphs

A graph is said to be Hamiltonian-connected if there exists a Hamiltonian path between any given pair of distinct vertices. In 1983, Thomassen proved that every 4-connected plane graph is Hamiltonian-connected, using the concept of Tutte subgraph. In this paper, we give a new proof to Thomassen’s theorem.