Path bipancyclicity of hypercubes

A bipartite graph is vertex-bipancyclic (respectively, edge-bipancyclic) if every vertex (respectively, edge) lies in a cycle of every even length from 4 to |V(G)| inclusive. It is easy to see that every connected edge-bipancyclic graph is vertex-bipancyclic. An n-dimensional hypercube, or n-cube denoted by Qn, is well known as bipartite and one of the most efficient networks for parallel computation. In this paper, we study a stronger bipancyclicity of hypercubes. We prove that every n-dimensional hypercube is (2n−4)-path-bipancyclic for n?3. That is, for any path P of length k with 1?k?2n−4 and any integer l with max{2,k}?l?²n−1, an even cycle C of length 2l can be found in Qn such that the path P is included in C for n?3.     

Embedding hamiltonian paths in hypercubes with a required vertex in a fixed position

Assume that n is a positive integer with n?2. It is proved that between any two different vertices x and y of Qn there exists a path Pl(x,y) of length l for any l with h(x,y)?l?n²−1 and 2|(l−h(x,y)). We expect such path Pl(x,y) can be further extended by including the vertices not in Pl(x,y) into a hamiltonian path from x to a fixed vertex z or a hamiltonian cycle. In this paper, we prove that for any two vertices x and z from different partite set of n-dimensional hypercube Qn, for any vertex y∈V(Qn)−{x,z}, and for any integer l with h(x,y)?l?n²−1−h(y,z) and 2|(l−h(x,y)), there exists a hamiltonian path R(x,y,z;l) from x to z such that dR(x,y,z;l)(x,y)=l. Moreover, for any two distinct vertices x and y of Qn and for any integer l with h(x,y)?l?²n−1 and 2|(l−h(x,y)), there exists a hamiltonian cycle S(x,y;l) such that dS(x,y;l)(x,y)=l.     

The panpositionable panconnectedness of augmented cubes

A graph G is panconnected if, for any two distinct vertices x and y of G, it contains an [x, y]-path of length l for each integer l satisfying d_G(x, y) ? l ? ∣V(G)∣ − 1, where d_G(x, y) denotes the distance between vertices x and y in G, and V(G) denotes the vertex set of G. For insight into the concept of panconnectedness, we propose a more refined property, namely panpositionable panconnectedness. Let x, y, and z be any three distinct vertices in a graph G. Then G is said to be panpositionably panconnected if for any d_G(x, z) ? l₁ ? ∣V(G)∣ − d_G(y, z) − 1, it contains a path P such that x is the beginning vertex of P, z is the (l₁ + 1)th vertex of P, and y is the (l₁ + l₂ + 1)th vertex of P for any integer l₂ satisfying d_G(y, z) ? l₂ ? ∣V(G)∣ − l₁ − 1. The augmented cube, proposed by Choudum and Sunitha [6] to be an enhancement of the n-cube Q_n, not only retains some attractive characteristics of Q_n but also possesses many distinguishing properties of which Q_n lacks. In this paper, we investigate the panpositionable panconnectedness with respect to the class of augmented cubes. As a consequence, many topological properties related to cycle and path embedding in augmented cubes, such as pancyclicity, panconnectedness, and panpositionable Hamiltonicity, can be drawn from our results.     

Edge-fault-tolerant diameter and bipanconnectivity of hypercubes

Let n(?3) be a given integer and . And let Qn be an n-dimensional hypercube and F⊂E(Qn), such that every vertex of the graph Qn−F is incident with at least two edges. Assume x and y are any two vertices with Hamming distance H(x,y)=h. In this paper, we obtain the following results: (1) If h?2 and |F|?min{n+h−1,2n−5}, then in Qn−F there exists an xy-path of each length l∈Ωh+2, and the upper bound n+h−1 on |F| is sharp when 2?h?n−4, and the upper bound 2n−5 on |F| is sharp when n−4?h?n−1 and h=2. (2) If |F|?2n−5, then in Qn−F there exists an xy-path of each length l∈Ωs, where s=h if n−1?h?n, and s=h+2 if n−4?h?n−2 and h?2, and s=h+4 otherwise. Hence, the diameter of the graph Qn−F is n. Our results improve some previous results.     

Hamiltonian paths and cycles passing through a prescribed path in hypercubes

Assume that P is any path in a bipartite graph G of length k with 2?k?h, G is said to be h-path bipancyclic if there exists a cycle C in G of every even length from 2k to |V(G)| such that P lies in C. In this paper, the following result is obtained: The n-dimensional hypercube Qn with n?3 is (2n−3)-path bipancyclic but is not (2n−2)-path bipancyclic, moreover, a path P of length k with 2?k?2n−3 lies in a cycle of length 2k−2 if and only if P contains two edges of the same dimension. In order to prove the above result we first show that any path of length at most 2n−1 is a subpath of a Hamiltonian path in Qn with n?2, moreover, the upper bound 2n−1 is sharp when n?4.     

Cycles passing through a prescribed path in a hypercube with faulty edges

Let Qn denote an n-dimensional hypercube with n?2, P be a path of length h in Qn and F⊂E(Qn)\E(P). Recently, Tsai proved that if 1?h?n−1 and |F|?n−1−h, then in the graph Qn−F the path P lies on a cycle of every even length from 2h+2 to n², and P also lies on a cycle of length 2h if |F|?h−2. In this paper, we show that if 1?h?2n−3 and |F|?n−2−⌊h/2⌋, then in Qn−F the path P lies on a cycle of every even length from 2h+2 to n², and P also lies on a cycle of length 2h if P contains two edges of the same dimension or P is a shortest path and |F∩E(Qh)|?h−2, where Qh is the h-dimensional subcube containing the path P. Moreover, the upper bound 2n−3 of h is sharp and the upper bound n−2−⌊h/2⌋ of |F| is sharp for any given h with 1?h?2n−3.     

Geodesic pancyclicity and balanced pancyclicity of Augmented cubes

For two distinct vertices u,v∈V(G), a cycle is called geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle; and a cycle C is called balanced cycle with u and v if dC(u,v)=max{dC(x,y)|x,y∈V(C)}. A graph G is pancyclic [J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and applications (1982) 271-278] if it contains a cycle of every length from 3 to |V(G)| inclusive. A graph G is called geodesic pancyclic [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181-187] (respectively, balanced pancyclic) if for each pair of vertices u,v∈V(G), it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max{2dG(u,v),3}?l?|V(G)|. Lai et al. [P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249-1251] proved that the n-dimensional Augmented cube, AQn, is pancyclic in the sense that a cycle of length l exists, 3?l?|V(AQn)|. In this paper, we study two new pancyclic properties and show that AQn is geodesic pancyclic and balanced pancyclic for n?2.     

Edge-bipancyclicity and edge-fault-tolerant bipancyclicity of bubble-sort graphs

A bipartite graph G is bipancyclic if G has a cycle of length l for every even 4?l?|V(G)|. For a bipancyclic graph G and any edge e, G is edge-bipancyclic if e lies on a cycle of any even length l of G. In this paper, we show that the bubble-sort graph Bn is bipancyclic for n?4 and also show that it is edge-bipancyclic for n?5. Assume that F is a subset of E(Bn). We prove that Bn−F is bipancyclic, when n?4 and |F|?n−3. Since Bn is a (n−1)-regular graph, this result is optimal in the worst case.     

On path bipancyclicity of hypercubes

Assume that P is any path in a bipartite graph G of length k with 2?k?h, G is said to be h-path bipancyclic if there exists a cycle C in G of every even length from 2k to |V(G)| such that P lies in C. Based on Lemma 5, the authors of [C.-H. Tsai, S.-Y. Jiang, Path bipancyclicity of hypercubes, Inform. Process. Lett. 101 (2007) 93-97] showed that the n-cube Qn with n?3 is (2n−4)-path bipancyclicity. In this paper, counterexamples to the lemma are given, therefore, their proof fails. And we show the following result: The n-cube Qn with n?3 is (2n−4)-path bipancyclicity but is not (2n−2)-path bipancyclicity, moreover, and a path P of length k with 2?k?2n−4 lies in a cycle of length 2k−2 if and only if P contains two edges of dimension i for some i, 1?i?n. We conjecture that if 2n−4 is replaced by 2n−3, then the above result also holds.     

Adjacent vertex-distinguishing edge and total chromatic numbers of hypercubes

An adjacent vertex-distinguishing edge coloring of a simple graph G is a proper edge coloring of G such that incident edge sets of any two adjacent vertices are assigned different sets of colors. A total coloring of a graph G is a coloring of both the edges and the vertices. A total coloring is proper if no two adjacent or incident elements receive the same color. An adjacent vertex-distinguishing total coloring h of a simple graph G=(V,E) is a proper total coloring of G such that H(u)≠H(v) for any two adjacent vertices u and v, where H(u)={h(wu)|wu∈E(G)}∪{h(u)} and H(v)={h(xv)|xv∈E(G)}∪{h(v)}. The minimum number of colors required for an adjacent vertex-distinguishing edge coloring (resp. an adjacent vertex-distinguishing total coloring) of G is called the adjacent vertex-distinguishing edge chromatic number (resp. adjacent vertex-distinguishing total chromatic number) of G and denoted by (resp. χat(G)). In this paper, we consider the adjacent vertex-distinguishing edge chromatic number and adjacent vertex-distinguishing total chromatic number of the hypercube Qn, prove that for n?3 and χat(Qn)=Δ(Qn)+2 for n?2.     

Geometric pattern matching for point sets in the plane under similarity transformations

We consider the following geometric pattern matching problem: Given two sets of points in the plane, P and Q, and some (arbitrary) δ>0, find a similarity transformation T (translation, rotation and scale) such that h(T(P),Q)<δ, where h(⋅,⋅) is the directional Hausdorff distance with L_∞ as the underlying metric; or report that none exists. We are only interested in the decision problem, not in minimizing the Hausdorff distance, since in the real world, where our applications come from, δ is determined by the practical uncertainty in the position of the points (pixels). Similarity transformations have not been dealt with in the context of the Hausdorff distance and we fill the gap here. We present efficient algorithms for this problem imposing a reasonable separation restriction on the points in the set Q. If the L_∞ distance between every pair of points in Q is at least 8δ, then the problem can be solved in O(mn²logn) time, where m and n are the numbers of points in P and Q respectively. If the L_∞ distance between every pair of points in Q is at least cδ, for some c, 0<c<1, we present a randomized approximate solution with expected runtime O(n²c−4ε−8log⁴mn), where ε>0 controls the approximation. Our approximation is on the size of the subset, B⊆P, such that h(T(B),Q)<δ and |B|>(1−ε)|P| with high probability.     

On edge connectivity of direct products of graphs

Let λ(G) be the edge connectivity of G. The direct product of graphs G and H is the graph with vertex set V(G×H)=V(G)×V(H), where two vertices (u₁,v₁) and (u₂,v₂) are adjacent in G×H if u₁u₂∈E(G) and v₁v₂∈E(H). We prove that λ(G×Kn)=min{n(n−1)λ(G),(n−1)δ(G)} for every nontrivial graph G and n?3. We also prove that for almost every pair of graphs G and H with n vertices and edge probability p, G×H is k-connected, where k=O(²(n/logn)).     

Exact exponential-time algorithms for finding bicliques

Due to a large number of applications, bicliques of graphs have been widely considered in the literature. This paper focuses on non-induced bicliques. Given a graph G=(V,E) on n vertices, a pair (X,Y), with X,Y⊆V, X∩Y=∅, is a non-induced biclique if {x,y}∈E for any x∈X and y∈Y. The NP-complete problem of finding a non-induced (k₁,k₂)-biclique asks to decide whether G contains a non-induced biclique (X,Y) such that |X|=k₁ and |Y|=k₂. In this paper, we design a polynomial-space O(n^1.6914)-time algorithm for this problem. It is based on an algorithm for bipartite graphs that runs in time O(n^1.30052). In deriving this algorithm, we also exhibit a relation to the spare allocation problem known from memory chip fabrication. As a byproduct, we show that the constraint bipartite vertex cover problem can be solved in time O(n^1.30052).     

Improved upper bounds on the L(2,1) -labeling of the skew and converse skew product graphs

An L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to the set of all nonnegative integers such that |f(x)−f(y)|≥2 if d(x,y)=1 and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between x and y in G. The L(2,1)-labeling number λ(G) of G is the smallest number k such that G has an L(2,1)-labeling with max{f(v):vV(G)}=k. Griggs and Yeh conjecture that λ(G)≤Δ² for any simple graph with maximum degree Δ≥2. This paper considers the graph formed by the skew product and the converse skew product of two graphs with a new approach on the analysis of adjacency matrices of the graphs as in [W.C. Shiu, Z. Shao, K.K. Poon, D. Zhang, A new approach to the L(2,1)-labeling of some products of graphs, IEEE Trans. Circuits Syst. II: Express Briefs (to appear)] and improves the previous upper bounds significantly.     

A Linear Time Algorithm for L(2,1)-Labeling of Trees

An L(2,1)-labeling of a graph G is an assignment f from the vertex set V(G) to the set of nonnegative integers such that |f(x)?f(y)|≥2 if x and y are adjacent and |f(x)?f(y)|≥1 if x and y are at distance 2, for all x and y in V(G). A k-L(2,1)-labeling is an L(2,1)-labeling f:V(G)→{0,…,k}, and the L(2,1)-labeling problem asks the minimum k, which we denote by λ(G), among all possible assignments. It is known that this problem is NP-hard even for graphs of treewidth 2, and tree is one of very few classes for which the problem is polynomially solvable. The running time of the best known algorithm for trees had been O(Δ ^4.5 n) for more than a decade, and an O(min{n ^1.75,Δ ^1.5 n})-time algorithm has appeared recently, where Δ and n are the maximum degree and the number of vertices of an input tree, however, it has been open if it is solvable in linear time. In this paper, we finally settle this problem by establishing a linear time algorithm for L(2,1)-labeling of trees. Furthermore, we show that it can be extended to a linear time algorithm for L(p,1)-labeling with a constant p.     

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

Hamiltonian properties of twisted hypercube-like networks with more faulty elements

Twisted hypercube-like networks (THLNs) are a large class of network topologies, which subsume some well-known hypercube variants. This paper is concerned with the longest cycle in an n-dimensional (n-D) THLN with up to 2n−9 faulty elements. Let G be an n-D THLN, n≥7. Let F be a subset of V(G)?E(G), |F|≤2n−9. We prove that G−F contains a Hamiltonian cycle if δ(G−F)≥2, and G−F contains a near Hamiltonian cycle if δ(G−F)≤1. Our work extends some previously known results.     

A fault-free Hamiltonian cycle passing through prescribed edges in a hypercube with faulty edges

In this paper, we consider the problem of a fault-free Hamiltonian cycle passing through prescribed edges in an n-dimensional hypercube Qn with some faulty edges. We obtain the following result: Let n?2, F⊂E(Qn), E₀⊂E(Qn)\F with 1?|E₀|?2n−3, |F|<n−(⌊|E₀|/2⌋+1). If the subgraph induced by E₀ is a linear forest (i.e., pairwise vertex-disjoint paths), then in the graph Qn−F all edges of E₀ lie on a Hamiltonian cycle.     

Edge-fault-tolerant edge-bipancyclicity of hypercubes

In this paper, we consider the problem embedding a cycle into the hypercube Qn with existence of faulty edges and show that for any edge subset F of Qn with |F|?n−1 every edge of Qn−F lies on a cycle of every even length from 6 to n² inclusive provided n?4 and all edges in F are not incident with the same vertex. This result improves some known results.     

Edge-pancyclicity and path-embeddability of bijective connection graphs

An n-dimensional Bijective Connection graph (in brief BC graph) is a regular graph with 2ⁿ nodes and n2ⁿ⁻¹ edges. The n-dimensional hypercube, crossed cube, Möbius cube, etc. are some examples of the n-dimensional BC graphs. In this paper, we propose a general method to study the edge-pancyclicity and path-embeddability of the BC graphs. First, we prove that a path of length l with dist(X_n, x, y) + 2 ? l ? 2ⁿ − 1 can be embedded between x and y with dilation 1 in X_n for x, y ∈ V(X_n) with x ≠ y in X_n, where X_n (n ? 4) is a n-dimensional BC graph satisfying the three specific conditions and V(X_n) is the node set of X_n. Furthermore, by this result, we can claim that X_n is edge-pancyclic. Lastly, we show that these results can be applied to not only crossed cubes and Möbius cubes, but also other BC graphs except crossed cubes and Möbius cubes. So far, the research on edge-pancyclicity and path-embeddability has been limited in some specific interconnection architectures such as crossed cubes, Möbius cubes.