Hamiltonian Bipartite Graph Research Articles

On the induced matching problem in hamiltonian bipartite graphs

Abstract In this paper, we study the parameterized complexity of the induced matching problem in hamiltonian bipartite graphs and the inapproximability of the maximum induced matching problem in hamiltonian bipartite graphs. We show that, given a hamiltonian bipartite graph, the induced matching problem is W[1]-hard and cannot be solved in time n o ⁢ ( k ) {n^{o(\sqrt{k})}} , where n is the number of vertices in the graph, unless the 3SAT problem can be solved in subexponential time. In addition, we show that unless NP = P {\operatorname{NP}=\operatorname{P}} , a maximum induced matching in a hamiltonian bipartite graph cannot be approximated within a ratio of n 1 / 4 - ϵ {n^{1/4-\epsilon}} , where n is the number of vertices in the graph.

A characterization of cycle-forced bipartite graphs

A forced cycleC of a graph G is a cycle in G such that G−V(C) has a unique perfect matching. A graph G is a cycle-forced graph if every cycle in G is a forced cycle. In this paper, we give a characterization of cycle-forced hamiltonian bipartite graphs.

Path Embeddings with Prescribed Edge in the Balanced Hypercube Network

The balanced hypercube network, which is a novel interconnection network for parallel computation and data processing, is a newly-invented variant of the hypercube. The particular feature of the balanced hypercube is that each processor has its own backup processor and they are connected to the same neighbors. A Hamiltonian bipartite graph with bipartition V 0 ∪ V 1 x ∈ V 0 y ∈ V 1 . It is known that each edge is on a Hamiltonian cycle of the balanced hypercube. In this paper, we prove that, for an arbitrary edge e in the balanced hypercube, there exists a Hamiltonian path between any two vertices x and y in different partite sets passing through e with e ≠ x y . This result improves some known results.

Edge Fault-Tolerant Strong Hamiltonian Laceability of Balanced Hypercubes

The balanced hypercube BHn, proposed by Wu and Huang, is a new variation of hypercube. A Hamiltonian bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path between two arbitrary vertices from different partite sets. A Hamiltonian laceable graph G is strongly Hamiltonian laceable if there is a path of length [Formula: see text] between any two distinct vertices of the same partite set. A graph G is called k-edge-fault strong Hamiltonian laceable, if G – F is strong Hamiltonian laceable for any edge-fault set F with [Formula: see text]. It has been proved that the balanced hypercube BHn is strong Hamiltonian laceable. In this paper, we improve the above result and prove that BHn is (n – 1)-edge-fault strong Hamiltonian laceable.

A characterization of PM-compact Hamiltonian bipartite graphs

The perfect matching polytope of a graph G is the convex hull of the incidence vectors of all perfect matchings in G. A graph is called perfect matching compact (shortly, PM-compact), if its perfect matching polytope has diameter one. This paper gives a complete characterization of simple PM-compact Hamiltonian bipartite graphs. We first define two families of graphs, called the H2C-bipartite graphs and the H23-bipartite graphs, respectively. Then we show that, for a simple Hamiltonian bipartite graph G with |V(G)| ≥ 6, G is PM-compact if and only if G is K3,3, or G is a spanning Hamiltonian subgraph of either an H2C-bipartite graph or an H23-bipartite graph.

Fault-tolerant Hamiltonian laceability of balanced hypercubes

The balanced hypercube, as a new variant of hypercube, has many desirable properties such as strong connectivity, high regularity and symmetry. The particular property of the balanced hypercube is that each processor has a backup processor sharing the same neighborhood. A Hamiltonian bipartite graph G=(V0∪V1,E) is said to be Hamiltonian laceable if there is a Hamiltonian path between any two vertices x∈V0 and y∈V1. It has been proved that the balanced hypercube BHn is Hamiltonian laceable for all n⩾1. In this paper, we have proved that after at most 2n-2 faulty edges occur, BHn remains Hamiltonian laceable for all n⩾2, this result is optimal with respect to the number of faulty edges can be tolerated in BHn.

Hyper-Hamiltonian laceability of balanced hypercubes

The balanced hypercube, proposed by Wu and Huang, is a new variation of hypercube. The particular property of the balanced hypercube is that each processor has a backup processor that shares the same neighborhood. A Hamiltonian bipartite graph with bipartition $$V_{0}\cup V_{1}$$ V 0 Â? V 1 is said to be Hamiltonian laceable if there is a Hamiltonian path between any two vertices $$x\in V_{0}$$ x Â? V 0 and $$y\in V_{1}$$ y Â? V 1 . A graph $$G$$ G is hyper-Hamiltonian laceable if it is Hamiltonian laceable and, for any vertex $$v\in V_{i}$$ v Â? V i , $$i\in \{0,1\}$$ i Â? { 0 , 1 } , there is a Hamiltonian path in G---v between any pair of vertices in $$V_{1-i}$$ V 1 - i . In this paper, we mainly prove that the balanced hypercube is hyper-Hamiltonian laceable.

The Degree and Regularity of Vanishing Ideals of Algebraic Toric Sets Over Finite Fields

Let X* be a subset of an affine space 𝔸 s , over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x → [x] and x → [(x, 1)], respectively, where [x] and [(x, 1)] are points in the projective spaces ℙ s−1 and ℙ s , respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud–Goto regularity conjecture. We give optimal bounds for the regularity when the graph is bipartite. It is shown that X* is an affine torus if and only if I(Y) is a complete intersection. We present some applications to coding theory and show some bounds for the minimum distance of parameterized linear codes for connected bipartite graphs.

Ring embedding in faulty generalized honeycomb torus – GHT(m, n, n/2)

The honeycomb torus HT(m) is an attractive architecture for distributed processing applications. For analysing its performance, a symmetric generalized honeycomb torus, GHT(m, n, n/2), with m≥2 and even n≥4, where m+n/2 is even, which is a 3-regular, Hamiltonian bipartite graph, is operated as a platform for combinatorial studies. More specifically, GHT(m, n, n/2) includes GHT(m, 6m, 3m), the isomorphism of the honeycomb torus HT(m). It has been proven that any GHT(m, n, n/2)−e is Hamiltonian for any edge e∈E(GHT(m, n, n/2)). Moreover, any GHT(m, n, n/2)−F is Hamiltonian for any F={u, v} with u∈B and v∈W, where B and W are the bipartition of V(GHT(m, n, n/2)) if and only if n≥6 or m=2, n≥4.

The bipanpositionable bipancyclic property of the hypercube

A bipartite graph is bipancyclic if it contains a cycle of every even length from 4 to |V(G)| inclusive. A hamiltonian bipartite graph G is bipanpositionable if, for any two different vertices x and y, there exists a hamiltonian cycle C of G such that dC(x,y)=k for any integer k with dG(x,y)≤k≤|V(G)|/2 and (k−dG(x,y)) being even. A bipartite graph G is k-cycle bipanpositionable if, for any two different vertices x and y, there exists a cycle of G with dC(x,y)=l and |V(C)|=k for any integer l with dG(x,y)≤l≤k2 and (l−dG(x,y)) being even. A bipartite graph G is bipanpositionable bipancyclic if G is k-cycle bipanpositionable for every even integer k, 4≤k≤|V(G)|. We prove that the hypercube Qn is bipanpositionable bipancyclic for n≥2.

Disjoint hamiltonian cycles in bipartite graphs

Let G = ( X , Y ) be a bipartite graph and define σ 2 2 ( G ) = min { d ( x ) + d ( y ) : x y ∉ E ( G ) , x ∈ X , y ∈ Y } . Moon and Moser [J. Moon, L. Moser, On Hamiltonian bipartite graphs, Israel J. Math. 1 (1963) 163–165. MR 28 # 4540] showed that if G is a bipartite graph on 2 n vertices such that σ 2 2 ( G ) ≥ n + 1 , then G is hamiltonian, sharpening a classical result of Ore [O. Ore, A note on Hamilton circuits, Amer. Math. Monthly 67 (1960) 55] for bipartite graphs. Here we prove that if G is a bipartite graph on 2 n vertices such that σ 2 2 ( G ) ≥ n + 2 k − 1 , then G contains k edge-disjoint hamiltonian cycles. This extends the result of Moon and Moser and a result of R. Faudree et al. [R. Faudree, C. Rousseau, R. Schelp, Edge-disjoint Hamiltonian cycles, Graph Theory Appl. Algorithms Comput. Sci. (1984) 231–249].

Globally bi- [formula omitted]-connected graphs

A k - container C ( x , y ) in a graph G = ( V , E ) is a set of k internally node-disjoint paths between vertices x and y . A k ∗ - container C ( x , y ) of G is a k -container such that every vertex of G is incident with a certain path in C ( x , y ) . A bipartite graph G = ( B ∪ W , E ) is globally bi- 3 ∗ - connected if there is a 3 ∗ -container C ( x , y ) between any pair of vertices { x , y } with x ∈ B and y ∈ W . Furthermore, G is hyper globally bi- 3 ∗ - connected if it is globally bi- 3 ∗ -connected and there exists a 3 ∗ -container C ( x , y ) in G − { z } for any three different vertices x , y , and z of the same partite set of G . A graph G = ( V , E ) is 1-edge Hamiltonian if G − e is Hamiltonian for any e ∈ E . A bipartite graph G = ( B ∪ W , E ) is 1 p - Hamiltonian if G − { x , y } is Hamiltonian for any pair of vertices { x , y } with x ∈ B and y ∈ W . In this paper, we prove that every hyper globally bi- 3 ∗ -connected graph is 1 p -Hamiltonian and every globally bi- 3 ∗ -connected graph is 1-edge Hamiltonian. We present some examples of hyper globally bi- 3 ∗ -connected graphs, some globally bi- 3 ∗ -connected graphs that are not hyper globally bi- 3 ∗ -connected, and some examples of 1-edge Hamiltonian bipartite graphs that are not globally bi- 3 ∗ -connected.

Pancyclism and Bipancyclism of Hamiltonian Graphs

In this paper, we prove the following two theorems: (1) If G is a hamiltonian graph of order n and if there exists a vertex x ∈ V(G) such that d(x) + d(y) ≥ n for each y not adjacent to x, then G is either pancyclic or the complete bipartite graph K(n/2, n/2). (2) Let G=(X, Y; E) be a hamiltonian bipartite graph with |X| = |Y| = n > 3. If there exists a vertex x ∈ X such that d(x) + d(y) ≥ n + 1 for each y ∈ Y not adjacent to x, then G is bipancyclic. The bounds in the two theorems are best possible.

Applying a condition for a hamiltonian bipartite graph to be bipancyclic

We apply the following result of Amar (1991) to balanced bipartite graphs satisfying the degree conditions of Moon and Moser (1963) and of Chvatal (1972):If there exist two vertices x1and x2, the distance between which is two, on a hamiltonian cycle ofa bipartite graph G of order 2n, such thatd(x1) + d(x2) ⩾ n + 1,then G is bipancyclic, except in some special cases. Thus, we show that these conditions sufficient to ensure the existence of a hamiltonian cycle also imply bipancyclism, and obtain a new proof (and easier) of a theorem of Mitchem and Schmeichel (1982).Then we give a new condition for a connected balanced bipartite graph of order 2n to be hamiltonian: dist(x,y) = 3 ⇒ d(x) + d(y) ⩾ n + 1, and, applying again the theorem above, we prove that this condition is sufficient for the graph to be bipancyclic.

A condition for a hamiltonian bipartite graph to be bipancyclic

Let G be a hamiltonian bipartite graph of order 2n and let C = (x>1,y1,x2,y2,…,xn,yn,x1) be a hamiltonian cycle of G. G is said to be bipancyclic if it contains a cycle of length 2l, for every l, 2⩽l⩽n. Suppose the vertices x1 and x2 are such that d(x1)+d(x2)⩾n>+1. Then G is either: 1.(1) bipancyclic,2.(2) missing a 4-cycle (then n is odd and the structure of G is known),3.(3) missing a (n + 1)-cycle (then n is odd and the structure of G is known).

On Hamilton's ideals

A theorem is proved that is, in a sense to be made precise, the best possible generalization of the theorems of Dirac, Pósa, and Bondy that give successively weaker sufficient conditions for a graph to be Hamiltonian. Some simple corollaries are deduced concerning Hamiltonian paths, n-Hamiltonian graphs, and Hamiltonian bipartite graphs.