Hamiltonian Cycle Research Articles

A New Approach to Determine the Minimal Polynomials of Binary Modified de Bruijn Sequences

A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique least-degree characteristic polynomial. Leveraging a recent characterization, we devise a novel general approach to determine the minimal polynomial. We translate the characterization into a problem of identifying a Hamiltonian cycle in a specially constructed graph. The graph is isomorphic to the modified de Bruijn–Good graph. Along the way, we demonstrate the usefulness of some computational tools from the cycle joining method in the modified setup.

Hamiltonian and long paths in bipartite graphs with connectivity

Let G be a graph, ν(G) the order of G, κ(G) the connectivity of G and k a positive integer such that k≤(ν(G)−2)/2. Then G is said to be k-extendable if it has a matching of size k and every matching of size k extends to a perfect matching of G. A Hamiltonian path of a graph G is a spanning path of G. A bipartite graph G with vertex sets V1 and V2 is defined to be Hamiltonian-laceable if such that |V1|=|V2| and for every pair of vertices p∈V1 and q∈V2, there exists a Hamiltonian path in G with endpoints p and q, or |V1|=|V2|+1 and for every pair of vertices p,q∈V1,p≠q, there exists a Hamiltonian path in G with endpoints p and q. Let G be a bipartite graph with bipartition (X,Y). Define bn(G) to be a maximum integer such that 0≤bn(G)<min{|X|,|Y|} and (1) for each non-empty subset S of X, if |S|≤|X|−bn(G), then |N(S)|≥|S|+bn(G), and if |X|−bn(G)<|S|≤|X|, then N(S)=Y; and (2) for each non-empty subset S of Y, if |S|≤|Y|−bn(G), then |N(S)|≥|S|+bn(G), and if |Y|−bn(G)<|S|≤|Y|, then N(S)=X; and (3) bn(G)=0 if there is no non-negative integer satisfying (1) and (2).Let G be a bipartite graph with bipartition (X,Y) such that |X|=|Y| and bn(G)>0. In this paper, we show that if ν(G)≤2κ(G)+4bn(G)−4, then G is Hamiltonian-laceable; or if ν(G)>6bn(G)−2, then for every pair of vertices x∈X and y∈Y, there is an (x,y)-path P in G with |V(P)|≥6bn(G)−2. We show some of its corollaries in k-extendable, bipartite graphs and a conjecture in k-extendable graphs.

Hamiltonian paths, unit-interval complexes, and determinantal facet ideals

We study d-dimensional generalizations of three mutually related topics in graph theory: Hamiltonian paths, (unit) interval graphs, and binomial edge ideals. We provide partial high-dimensional generalizations of Ore and Pósa's sufficient conditions for a graph to be Hamiltonian. We introduce a hierarchy of combinatorial properties for simplicial complexes that generalize unit-interval, interval, and co-comparability graphs. We connect these properties to the already existing notions of determinantal facet ideals and (tight and weak) Hamiltonian paths in simplicial complexes. Some important consequences of our work are:(1)Every unit-interval strongly-connected d-dimensional simplicial complex is traceable.(This extends the well-known result “unit-interval connected graphs are traceable”.)(2)Every unit-interval d-complex that remains strongly connected after the deletion of d or less vertices, is Hamiltonian.(This extends the fact that “unit-interval 2-connected graphs are Hamiltonian”.)(3)Unit-interval complexes are characterized, among traceable complexes, by the property that the minors defining their determinantal facet ideal form a Gröbner basis for a diagonal term order which is compatible with the traceability of the complex.(This corrects a recent theorem by Ene et al., extends a result by Herzog and others, and partially answers a question by Almousa–Vandebogert.)(4)Only the d-skeleton of the simplex has a determinantal facet ideal with linear resolution.(This extends the result by Kiani and Saeedi-Madani that “only the complete graph has a binomial edge ideal with linear resolution”.)(5)The determinantal facet ideals of all under-closed and semi-closed complexes have a square-free initial ideal with respect to lex. In characteristic p, they are even F-pure.

On the hamiltonicity of a planar graph and its vertex‐deleted subgraphs

AbstractTutte proved that every planar 4‐connected graph is hamiltonian. Thomassen showed that the same conclusion holds for the superclass of planar graphs with minimum degree at least 4 in which all vertex‐deleted subgraphs are hamiltonian. We here prove that if in a planar ‐vertex graph with minimum degree at least 4 at least vertex‐deleted subgraphs are hamiltonian, then the graph contains two hamiltonian cycles, but that for every there exists a nonhamiltonian polyhedral ‐vertex graph with minimum degree at least 4 containing hamiltonian vertex‐deleted subgraphs. Furthermore, we study the hamiltonicity of planar triangulations and their vertex‐deleted subgraphs as well as Bondy's meta‐conjecture, and prove that a polyhedral graph with minimum degree at least 4 in which all vertex‐deleted subgraphs are traceable, must itself be traceable.

Hamiltonian tours in color

Hamiltonian tours in color

A note on hamiltonian cycles in 4-tough (P2 ∪ kP1)-free graphs

Let t>0 be a real number and let G be a graph. We say G is t-tough if for every cutset S of G, the ratio of |S| to the number of components of G−S is at least t. The Toughness Conjecture of Chvátal, stating that there exists a constant t0 such that every t0-tough graph with at least three vertices is hamiltonian, is still open in general. For any given integer k≥1, a graph G is (P2∪kP1) free if G does not contain the disjoint union of P2 and k isolated vertices as an induced subgraph. In this note, we show that every 4-tough and 2k-connected (P2∪kP1)-free graph with at least three vertices is hamiltonian. This result in some sense is an “extension” of the classical Chvátal-Erdős Theorem that every max⁡{2,k}-connected (k+1)P1-free graph on at least three vertices is hamiltonian.

Hamiltonian Paths and Cycles in Some 4-Uniform Hypergraphs

In 1999, Katona and Kierstead conjectured that if a k-uniform hypergraph \({\mathcal {H}}\) on n vertices has minimum co-degree \(\lfloor \frac{n-k+3}{2}\rfloor \), i.e., each set of \(k-1\) vertices is contained in at least \(\lfloor \frac{n-k+3}{2}\rfloor \) edges, then it has a Hamiltonian cycle. Rödl, Ruciński and Szemerédi in 2011 proved that the conjecture is true when \(k=3\) and n is large. We show that this Katona-Kierstead conjecture holds if \(k=4\), n is large, and \(V({\mathcal {H}})\) has a partition A, B such that \(|A|=\lceil n/2\rceil \), \(|\{e\in E({\mathcal {H}}):|e \cap A|=2\}| <\epsilon n^{4}\) for a fixed small constant \(\epsilon >0\).

Two completely independent spanning trees of claw-free graphs

If a graph G contains two spanning trees T1,T2 such that for each two distinct vertices x,y of G, the (x,y)-path in each Ti has no common edge and no common vertex except for the two ends, then T1,T2 are called two completely independent spanning trees (CISTs) of G,i∈{1,2}. If the subgraph induced by the neighbor set of a vertex u in a graph G is connected, then u is an eligible vertex of G. An hourglass is a graph with degree 4,2,2,2,2 and (P6)2 denotes the square graph of a path P6 on six vertices. Araki (2014) [1] proved that if a graph G satisfies the Dirac's condition, which is also sufficient condition for hamiltonian graphs, then G contains two CISTs. Ryjáček proposed that a claw-free graph G is hamiltonian if and only if its Ryjáček's closure, denoted by cl(G), is hamiltonian. In this paper, we prove that if G is a {claw, hourglass, (P6)2}-free graph with δ(G)≥4, then G contains two CISTs if and only if cl(G) has two CISTs. The bound of the minimum degree in our result is best possible.

Geodesic bipancyclicity of the Cartesian product of graphs

A cycle containing a shortest path between two vertices $u$ and $v$ in a graph $G$ is called a $(u,v)$-geodesic cycle. A connected graph $G$ is geodesic 2-bipancyclic, if every pair of vertices $u,v$ of it is contained in a $(u,v)$-geodesic cycle of length $l$ for each even integer $l$ satisfying $2d + 2\leq l \leq |V(G)|,$ where $d$ is the distance between $u$ and $v.$ In this paper, we prove that the Cartesian product of two geodesic hamiltonian graphs is a geodesic 2-bipancyclic graph. As a consequence, we show that for $n \geq 2$ every $n$-dimensional torus is a geodesic 2-bipancyclic graph.

The Hardest Hamiltonian Cycle Problem Instances: The Plateau of Yes and the Cliff of No

We use two evolutionary algorithms to make hard instances of the Hamiltonian cycle problem. Hardness (or ‘fitness’), is defined as the number of recursions required by Vandegriend–Culberson, the best known exact backtracking algorithm for the problem. The hardest instances, all non-Hamiltonian, display a high degree of regularity and scalability across graph sizes. These graphs are found multiple times through independent runs, and by both evolutionary algorithms, suggesting the search space might contain monotonic paths towards the global maximum. For Hamiltonian-bound evolution, some hard graphs were found, but convergence is much less consistent. In this extended paper, we survey the neighbourhoods of both the hardest yes- and no-instances produced by the evolutionary algorithms. Results show that the hardest no-instance resides on top of a steep cliff, while the hardest yes-instance turns out to be part of a plateau of 27 equally hard instances. While definitive answers are far away, the results provide a lot of insight in the Hamiltonian cycle problem’s state space.

The Circlet Inequalities: A New, Circulant-Based, Facet-Defining Inequality for the TSP

Facet-defining inequalities of the symmetric traveling salesman problem (TSP) polytope play a prominent role in both polyhedral TSP research and state-of-the-art TSP solvers. In this paper, we introduce a new class of facet-defining inequalities, the circlet inequalities. These inequalities were first conjectured in Gutekunst and Williamson [Gutekunst SC, Williamson DP (2019) Characterizing the integrality gap of the subtour LP for the circulant traveling salesman problem. SIAM J. Discrete Math. 33(4):2452–2478] when studying the circulant TSP, and they provide a bridge between polyhedral TSP research and number-theoretic investigations of Hamiltonian cycles stemming from a conjecture from Marco Buratti in 2007. The circlet inequalities exhibit circulant symmetry by placing the same weight on all edges of a given length; our main proof exploits this symmetry to prove the validity of the circlet inequalities. We then show that the circlet inequalities are facet-defining and compute their strength following Goemans [Goemans MX (1995) Worst-case comparison of valid inequalities for the TSP. Math. Programming 69:335–349]; they achieve the same worst case strength as the similarly circulant crown inequalities of Naddef and Rinaldi [Naddef D, Rinaldi G (1992) The crown inequalities for the symmetric traveling salesman polytope. Math. Oper. Res. 17(2):308–326] but are generally stronger. Funding: This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program [Grant DGE-1650441] and by the National Science Foundation Division of Computing and Communications Foundations [Grant CCF-1908517].

Hamiltonian cycles in 7-tough (P3 ∪ 2P1)-free graphs

The toughness of a noncomplete graph G is the maximum real number t such that the ratio of |S| to the number of components of G−S is at least t for every cutset S of G. Determining the toughness for a given graph is NP-hard. Chvátal's toughness conjecture, stating that there exists a constant t0 such that every graph with toughness at least t0 is hamiltonian, is still open for general graphs. A graph is called (P3∪2P1)-free if it does not contain any induced subgraph isomorphic to P3∪2P1, the disjoint union of P3 and two isolated vertices. In this paper, we confirm Chvátal's toughness conjecture for (P3∪2P1)-free graphs by showing that every 7-tough (P3∪2P1)-free graph on at least three vertices is hamiltonian.

Spectral sufficient conditions of the existence of the longest path in the graph and its traceability

It is known that the existence of many classical combinatorial structures in a graph, such as perfect matchings, Hamiltonian cycles, effective dominating sets, etc., can be characterized using (κ,τ)-regular sets, whose determination is equivalent to the determination of these classical combinatorial structures. On the other hand, the determination of (κ,τ)regular sets is closely related to the properties of the main spectrum of a graph. We use the previously obtained generalized properties of (κ,τ)-regular sets of graphs to develop a recognition algorithm of the traceability of a graph. We also obtained new sufficient conditions for the existence of a longest simple path in a graph in terms of the spectral radius of the adjacency matrix and the signless Laplacian of the graph.

Rainbow matchings in an edge-colored planar bipartite graph

In this paper, we consider the existence of rainbow matchings in maximal bipartite planar graphs. We determine the maximum number of colors appearing in an edge-coloring of maximal bipartite planar graphs with a Hamilton cycle which does not contain any rainbow k-matching. The bounds given are best possible.

An Embedded Hamiltonian Graph-Guided Heuristic Algorithm for Two-Echelon Vehicle Routing Problem.

Two-echelon vehicle routing problem (2E-VRP) is an NP-hard combinatorial optimization problem and a basic mathematical model of modern city logistics. While it is difficult to obtain the optimal solution of 2E-VRP, this study finds a breakthrough that the structure of the optimal route planning for 2E-VRP is usually an embedded Hamiltonian graph. In the graph, routes can be drawn in a planar graph as Hamiltonian circuits without intersections. Based on this finding, an embedded Hamiltonian graph-guided heuristic algorithm is proposed to solve 2E-VRP. As a crucial part of the algorithm, an initialization scheme is designed to search for the farthest vertices from each route and insert the rest of the vertices. In the satellite-adjustment process, a dynamic adjustment for satellites scheme is proposed to adjust the state of satellites. The two schemes aim to construct Hamiltonian circuits with few intersections. Experiments have been conducted on 207 instances to demonstrate the effect of the proposed algorithm on solving 2E-VRP. Experimental results show that the proposed algorithm can obtain more solutions of 2E-VRP with significantly smaller objective-function values. Furthermore, the number of intersections in routes generated by the proposed algorithm is much less than those obtained by the compared algorithms. With the use of the two schemes, the embedded Hamiltonian graph-guided heuristic algorithm significantly outperforms the compared algorithms for 2E-VRP.

Hamiltonian Paths of -cubes Avoiding Faulty Links and Passing Through Prescribed Linear Forests

The <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -ary <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -cube <inline-formula><tex-math notation="LaTeX">$Q_n^k$</tex-math></inline-formula> is one of the most attractive interconnection networks for parallel and distributed systems. Let <inline-formula><tex-math notation="LaTeX">$F$</tex-math></inline-formula> be a set of faulty links in <inline-formula><tex-math notation="LaTeX">$Q_n^k$</tex-math></inline-formula> and let <inline-formula><tex-math notation="LaTeX">$L$</tex-math></inline-formula> be a linear forest in <inline-formula><tex-math notation="LaTeX">$Q_n^k-F$</tex-math></inline-formula> such that <inline-formula><tex-math notation="LaTeX">$|E(L)|+|F|\leq 2n-3$</tex-math></inline-formula> . For any two distinct nodes <inline-formula><tex-math notation="LaTeX">$u$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$v$</tex-math></inline-formula> of <inline-formula><tex-math notation="LaTeX">$Q_n^k$</tex-math></inline-formula> with <inline-formula><tex-math notation="LaTeX">$n\geq 2$</tex-math></inline-formula> and odd <inline-formula><tex-math notation="LaTeX">$k\geq 3$</tex-math></inline-formula> , we prove that <inline-formula><tex-math notation="LaTeX">$Q_n^k-F$</tex-math></inline-formula> admits a Hamiltonian path between <inline-formula><tex-math notation="LaTeX">$u$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$v$</tex-math></inline-formula> passing through <inline-formula><tex-math notation="LaTeX">$L$</tex-math></inline-formula> if and only if none of the paths in <inline-formula><tex-math notation="LaTeX">$L$</tex-math></inline-formula> has <inline-formula><tex-math notation="LaTeX">$u$</tex-math></inline-formula> or <inline-formula><tex-math notation="LaTeX">$v$</tex-math></inline-formula> as internal nodes or both of them as end-nodes. The upper bound <inline-formula><tex-math notation="LaTeX">$2n-3$</tex-math></inline-formula> on <inline-formula><tex-math notation="LaTeX">$|E(L)|+|F|$</tex-math></inline-formula> is optimal in the worst case. The main results in this paper generalized some known results.

Sufficient spectral conditions for graphs being k-edge-Hamiltonian or k-Hamiltonian

A graph G is k-edge-Hamiltonian if any collection of vertex-disjoint paths with at most k edges altogether belong to a Hamiltonian cycle in G. A graph G is k-Hamiltonian if for all with , the subgraph induced by has a Hamiltonian cycle. These two concepts are classical extensions of the usual Hamiltonian graphs. In this paper, we present some spectral sufficient conditions for a graph to be k-edge-Hamiltonian and k-Hamiltonian in terms of the adjacency spectral radius as well as the signless Laplacian spectral radius. Our results could be viewed as slight extensions of the recent theorems proved by Li and Ning [Linear Multilinear Algebra. 2016;64:2252–2269], Nikiforov [Czechoslovak Math J. 2016;66:925–940] and Li et al. [Linear Multilinear Algebra. 2018;66:2011–2023]. Moreover, we shall prove a stability result for graphs being k-Hamiltonian, which could be regarded as a complement of two recent results of Füredi et al. [Discrete Math. 2017;340:2688–2690] and [Discrete Math. 2019;342:1919–1923].

Connected Square Network Graphs

In this study, connected square network graphs are introduced and two different definitions are given. Firstly, connected square network graphs are shown to be a Hamilton graph. Further, the labelling algorithm of this graph is obtained by using gray code. Finally, its topological properties are obtained, and conclusion are given.

Structure and Complexity of 2-Intersection Graphs of 3-Hypergraphs

Given a 3-uniform hypergraph H having a set V of vertices, and a set of hyperedges Tsubset mathcal {P}(V), whose elements have cardinality three each, a null labelling is an assignment of pm 1 to the hyperedges such that each vertex belongs to the same number of hyperedges labelled +1 and -1. A sufficient condition for the existence of a null labelling of H (proved in Di Marco et al. Lect Notes Comput Sci 12757:282–294, 2021) is a Hamiltonian cycle in its 2-intersection graph. The notion of 2-intersection graph generalizes that of intersection graph of an (hyper)graph and extends its effectiveness. The present study first shows that this sufficient condition for the existence of a null labelling in H can not be weakened by requiring only the connectedness of the 2-intersection graph. Then some interesting properties related to their clique configurations are proved. Finally, the main result is proved, the NP-completeness of this characterization and, as a consequence, of the construction of the related 3-hypergraphs.

Hamiltonian cycles in 2‐tough 2K2 $2{K}_{2}$‐free graphs

AbstractA graph is called a ‐free graph if it does not contain as an induced subgraph. In 2014, Broersma, Patel, and Pyatkin showed that every 25‐tough ‐free graph on at least three vertices is Hamiltonian. Recently, Shan improved this result by showing that 3‐tough is sufficient instead of 25‐tough. In this paper, we show that every 2‐tough ‐free graph on at least three vertices is Hamiltonian, which was conjectured by Gao and Pasechnik.