The super laceability of the hypercubes

A k-containerC(u,v) of a graph G is a set of k disjoint paths joining u to v. A k-container C(u,v) is a k∗-container if every vertex of G is incident with a path in C(u,v). A bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u, v from different partite set of G. A bipartite graph G with connectivity k is super laceable if it is i∗-laceable for all i?k. A bipartite graph G with connectivity k is f-edge fault-tolerant super laceable if G−F is i∗-laceable for any 1?i?k−f and for any edge subset F with |F|=f<k−1. In this paper, we prove that the hypercube graph Q_r is super laceable. Moreover, Q_r is f-edge fault-tolerant super laceable for any f?r−2.     

On the 1-fault hamiltonicity for graphs satisfying Ore's theorem and its generalization

Consider any undirected and simple graph G=(V, E), where V and E denote the vertex set and the edge set of G, respectively. Let |G|=|V|=n. The well-known Ore's theorem states that if deg_G(u)+deg_G(v)≥n+k holds for each pair of nonadjacent vertices u and v of G, then G is traceable for k=?1, hamiltonian for k=0, and hamiltonian-connected for k=1. Lin et al. generalized Ore's theorem and showed that under the same condition as above, G is r*-connected for 1≤r≤k+2 with k≥1. In this paper, we improve both theorems by showing that the hamiltonicity or r*-connectivity of any graph G satisfying the condition deg_G(u)+deg_G(v)≥n+k with k≥?1 is preserved even after one vertex or one edge is removed, unless G belongs to two exceptional families.     

The super-connected property of recursive circulant graphs

In a graph G, a k-container C_k(u,v) is a set of k disjoint paths joining u and v. A k-container C_k(u,v) is k∗-container if every vertex of G is passed by some path in C_k(u,v). A graph G is k∗-connected if there exists a k∗-container between any two vertices. An m-regular graph G is super-connected if G is k∗-connected for any k with 1?k?m. In this paper, we prove that the recursive circulant graphs G(2^m,4), proposed by Park and Chwa [Theoret. Comput. Sci. 244 (2000) 35-62], are super-connected if and only if m≠2.     

Multiplicative vertex-colouring weightings of graphs

In this paper we conjecture that the edges of any non-trivial graph can be weighted with integers 1, 2, 3 in such a way that for every edge uv the product of weights of the edges adjacent to u is different than the product of weights of the edges adjacent to v. It is proven here for cycles, paths, complete graphs and 3-colourable graphs. It is also shown that the edges of every non-trivial graph can be weighted with integers 1, 2, 3, 4 in such a way that the adjacent vertices have different products of incident edge weights.In a total weighting of a simple graph G we assign the positive integers to edges and to vertices of G. We consider a colouring of G obtained by assigning to each vertex v the product of its weight and the weights of its adjacent edges. The paper conjectures that we can get the proper colouring in this way using the weights 1, 2 for every simple graph. We show that we can do it using the weights 1, 2, 4 on edges and 1, 2 on vertices.     

Approximation Algorithms for Minimum K -Cut

Let G=(V,E) be a complete undirected graph, with node set V={v ₁ , . . ., v _n } and edge set E . The edges (v _i ,v _j ) ∈ E have nonnegative weights that satisfy the triangle inequality. Given a set of integers K = { k _i } _i=1 ^p , the minimum K-cut problem is to compute disjoint subsets with sizes { k _i } _i=1 ^p , minimizing the total weight of edges whose two ends are in different subsets. We demonstrate that for any fixed p it is possible to obtain in polynomial time an approximation of at most three times the optimal value. We also prove bounds on the ratio between the weights of maximum and minimum cuts. Received September 4, 1997; revised July 15, 1998.     

A polynomial time algorithm for obtaining minimum edge ranking on two-connected outerplanar graphs

An edge ranking of a graph G is a labeling r of its edges with positive integers such that every path between two different edges eu, ev with the same rank r(eu)=r(ev) contains an intermediate edge ew with rank r(ew)>r(eu). An edge ranking of G is minimum if the largest rank k assigned is the smallest among all rankings of G. The edge ranking problem is to find a minimum edge ranking of given graph G. This problem is NP-hard and no polynomial time algorithm for solving it is known for non-trivial classes of graphs other than the class of trees. In this paper, we first show, on a general graph G, a relation between a minimum edge ranking of G and its minimal cuts, which ensures that we can obtain a polynomial time algorithm for obtaining minimum edge ranking of a given graph G if minimal cuts for each subgraph of G can be found in polynomial time and the number of those is polynomial. Based on this relation, we develop a polynomial time algorithm for finding a minimum edge ranking on a 2-connected outerplanar graph.     

Component connectivity of generalized Petersen graphs

Let G be a simple non-complete graph of order n. The r-component edge connectivity of G denoted as λ_r (G) is the minimum number of edges that must be removed from G in order to obtain a graph with (at least) r connected components. The concept of r-component edge connectivity generalizes that of edge connectivity by taking into account the number of components of the resulting graph. In this paper we establish bounds of the r component edge connectivity of an important family of interconnection network models, the generalized Petersen graphs GP(n, k) in which n and k are relatively prime integers.     

General-demand disjoint path covers in a graph with faulty elements

A k-disjoint path cover of a graph is a set of k internally vertex-disjoint paths which cover the vertex set with k paths and each of which runs between a source and a sink. Given that each source and sink v is associated with an integer-valued demand d(v)≥1, we are concerned with general-demand k-disjoint path cover in which every source and sink v is contained in the d(v) paths. In this paper, we present a reduction of a general-demand disjoint path cover problem to an unpaired many-to-many disjoint path cover problem, and obtain some results on disjoint path covers of restricted HL-graphs and proper interval graphs with faulty vertices and/or edges.     

An improved approximation algorithm for the metric maximum clustering problem with given cluster sizes

The input to the metric maximum clustering problem with given cluster sizes consists of a complete graph G=(V,E) with edge weights satisfying the triangle inequality, and integers c₁,…,cp. The goal is to find a partition of V into disjoint clusters of sizes c₁,…,cp, maximizing the sum of weights of edges whose two ends belong to the same cluster. We describe an approximation algorithms for this problem with performance guarantee that approaches 0.5 when the cluster sizes are large.     

The super spanning connectivity and super spanning laceability of the enhanced hypercubes

A k -container C(u,v) of a graph G is a set of k disjoint paths between u and v. A k-container C(u,v) of G is a k ^* -container if it contains all vertices of G. A graph G is k ^* -connected if there exists a k ^*-container between any two distinct vertices of G. Therefore, a graph is 1^*-connected (respectively, 2^*-connected) if and only if it is Hamiltonian connected (respectively, Hamiltonian). A graph G is super spanning connected if there exists a k ^*-container between any two distinct vertices of G for every k with 1≤k≤κ(G) where κ(G) is the connectivity of G. A bipartite graph G is k ^* -laceable if there exists a k ^*-container between any two vertices from different partite set of G. A bipartite graph G is super spanning laceable if there exists a k ^*-container between any two vertices from different partite set of G for every k with 1≤k≤κ(G). In this paper, we prove that the enhanced hypercube Q _n,m is super spanning laceable if m is an odd integer and super spanning connected if otherwise.

On identifying codes that are robust against edge changes

Assume that G = (V, E) is an undirected graph, and C ⊆ V. For every v ∈ V, denote I_r(G; v) = {u ∈ C: d(u,v) ≤ r}, where d(u,v) denotes the number of edges on any shortest path from u to v in G. If all the sets I_r(G; v) for v ∈ V are pairwise different, and none of them is the empty set, the code C is called r-identifying. The motivation for identifying codes comes, for instance, from finding faulty processors in multiprocessor systems or from location detection in emergency sensor networks. The underlying architecture is modelled by a graph. We study various types of identifying codes that are robust against six natural changes in the graph; known or unknown edge deletions, additions or both. Our focus is on the radius r = 1. We show that in the infinite square grid the optimal density of a 1-identifying code that is robust against one unknown edge deletion is 1/2 and the optimal density of a 1-identifying code that is robust against one unknown edge addition equals 3/4 in the infinite hexagonal mesh. Moreover, although it is shown that all six problems are in general different, we prove that in the binary hypercube there are cases where five of the six problems coincide.     

Not necessarily proper total colourings which are adjacent vertex distinguishing

Let G be a simple graph with no isolated edge. Let f be a total colouring of G which is not necessarily proper. f is said to be adjacent vertex distinguishing if for any pair of adjacent vertices u, v of G, we have C(u)≠C(v), where C(u) denotes the set of colours of u and its incident edges under f. The minimum number of colours required for an adjacent vertex distinguishing not necessarily proper total colouring of G is called the adjacent vertex distinguishing not necessarily proper total chromatic number. Seven kinds of adjacent vertex distinguishing not necessarily proper total colourings are introduced in this paper. Some results of adjacent vertex distinguishing not necessarily proper total chromatic numbers are obtained and some conjectures are also proposed.     

The spanning connectivity of folded hypercubes

A k-container of a graph G is a set of k internally disjoint paths between u and v. A k-container of G is a k∗-container if it contains all vertices of G. A graph G is k∗-connected if there exists a k∗-container between any two distinct vertices, and a bipartite graph G is k∗-laceable if there exists a k∗-container between any two vertices u and v from different partite sets of G for a given k. A k-connected graph (respectively, bipartite graph) G is f-edge fault-tolerant spanning connected (respectively, laceable) if G − F is w∗-connected for any w with 1 ? w ? k − f and for any set F of f faulty edges in G. This paper shows that the folded hypercube FQ_n is f-edge fault-tolerant spanning laceable if n(?3) is odd and f ? n − 1, and f-edge fault-tolerant spanning connected if n (?2) is even and f ? n − 2.     

Approximating Minimum-Power Degree and Connectivity Problems

Power optimization is a central issue in wireless network design. Given a graph with costs on the edges, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider several fundamental undirected network design problems under the power minimization criteria. Given a graph G=(V,E)\mathcal{G}=(V,\mathcal{E}) with edge costs {c(e):e∈ℰ} and degree requirements {r(v):v∈V}, the Minimum-Power Edge-Multi-Cover\textsf{Minimum-Power Edge-Multi-Cover} (MPEMC\textsf{MPEMC} ) problem is to find a minimum-power subgraph G of G\mathcal{G} so that the degree of every node v in G is at least r(v). We give an O(log n)-approximation algorithms for MPEMC\textsf{MPEMC} , improving the previous ratio O(log ⁴ n). This is used to derive an O(log n+α)-approximation algorithm for the undirected $\textsf{Minimum-Power $\textsf{Minimum-Power ($\textsf{MP$\textsf{MP ) problem, where α is the best known ratio for the min-cost variant of the problem. Currently, _boxclosen-k)\alpha=O(\log k\cdot \log\frac{n}{n-k}) which is O(log k) unless k=n−o(n), and is O(log ² k)=O(log ² n) for k=n−o(n). Our result shows that the min-power and the min-cost versions of the $\textsf{$\textsf{ problem are equivalent with respect to approximation, unless the min-cost variant admits an o(log n)-approximation, which seems to be out of reach at the moment.     

Fast Minor Testing in Planar Graphs

Minor Containment is a fundamental problem in Algorithmic Graph Theory used as a subroutine in numerous graph algorithms. A model of a graph H in a graph G is a set of disjoint connected subgraphs of G indexed by the vertices of H, such that if {u,v} is an edge of H, then there is an edge of G between components C _u and C _v . A graph H is a minor of G if G contains a model of H as a subgraph. We give an algorithm that, given a planar n-vertex graph G and an h-vertex graph H, either finds in time $\mathcal{O}(2^{\mathcal{O}(h)} \cdot n +n^{2}\cdot\log n)$ a model of H in G, or correctly concludes that G does not contain H as a minor. Our algorithm is the first single-exponential algorithm for this problem and improves all previous minor testing algorithms in planar graphs. Our technique is based on a novel approach called partially embedded dynamic programming.     

Radio number of ladder graphs

Let G be a connected graph with diameter diam(G). The radio number for G, denoted by rn(G), is the smallest integer k such that there exists a function f: V(G)→{0, 1, 2, …, k} with the following satisfied for all vertices u and v:|f(u)?f(v)|≥diam (G)?d _G (u, v)+1, where d _G (u, v) is the distance between u and v in G. In this paper, we determine the radio number of ladder graphs.     

Efficient algorithms for clique problems

The k-clique problem is a cornerstone of NP-completeness and parametrized complexity. When k is a fixed constant, the asymptotically fastest known algorithm for finding a k-clique in an n-node graph runs in O(n0.792k) time (given by Nešet?il and Poljak). However, this algorithm is infamously inapplicable, as it relies on Coppersmith and Winograd's fast matrix multiplication.We present good combinatorial algorithms for solving k-clique problems. These algorithms do not require large constants in their runtime, they can be readily implemented in any reasonable random access model, and are very space-efficient compared to their algebraic counterparts. Our results are the following:

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We give an algorithm for k-clique that runs in O(nk/(εlogn)^k−1) time and O(nε) space, for all ε>0, on graphs with n nodes. This is the first algorithm to take o(nk) time and O(nc) space for c independent of k.

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Let k be even. Define a k-semiclique to be a k-node graph G that can be divided into two disjoint subgraphs U={u₁,…,uk/2} and V={v₁,…,vk/2} such that U and V are cliques, and for all i?j, the graph G contains the edge {ui,vj}. We give an time algorithm for determining if a graph has a k-semiclique. This yields an approximation algorithm for k-clique, in the following sense: if a given graph contains a k-clique, then our algorithm returns a subgraph with at least 3/4 of the edges in a k-clique.

     

Hamiltonian connectivity and globally 3∗-connectivity of dual-cube extensive networks

In 2000, Li et al. introduced dual-cube networks, denoted by DC_n for n?1, using the hypercube family Q_n and showed the vertex symmetry and some fault-tolerant hamiltonian properties of DC_n. In this article, we introduce a new family of interconnection networks called dual-cube extensive networks, denoted by DCEN(G). Given any arbitrary graph G, DCEN(G) is generated from G using the similar structure of DC_n. We show that if G is a nonbipartite and hamiltonian connected graph, then DCEN(G) is hamiltonian connected. In addition, if G has the property that for any two distinct vertices u,v of G, there exist three disjoint paths between u and v such that these three paths span the graph G, then DCEN(G) preserves the same property. Furthermore, we prove that the similar results hold when G is a bipartite graph.     

The construction of mutually independent Hamiltonian cycles in bubble-sort graphs

A Hamiltonian cycle C=? u ₁, u ₂, …, u _n(G), u ₁ ? with n(G)=number of vertices of G, is a cycle C(u ₁; G), where u ₁ is the beginning and ending vertex and u _i is the ith vertex in C and u _i ≠u _j for any i≠j, 1≤i, j≤n(G). A set of Hamiltonian cycles {C ₁, C ₂, …, C _k } of G is mutually independent if any two different Hamiltonian cycles are independent. For a hamiltonian graph G, the mutually independent Hamiltonianicity number of G, denoted by h(G), is the maximum integer k such that for any vertex u of G there exist k-mutually independent Hamiltonian cycles of G starting at u. In this paper, we prove that h(B _n )=n?1 if n≥4, where B _n is the n-dimensional bubble-sort graph.     

On the oriented chromatic number of grids

In this paper, we focus on the oriented coloring of graphs. Oriented coloring is a coloring of the vertices of an oriented graph G without symmetric arcs such that (i) no two neighbors in G are assigned the same color, and (ii) if two vertices u and v such that (u,v)∈A(G) are assigned colors c(u) and c(v), then for any (z,t)∈A(G), we cannot have simultaneously c(z)=c(v) and c(t)=c(u). The oriented chromatic number of an unoriented graph G is the smallest number k of colors for which any of the orientations of G can be colored with k colors.The main results we obtain in this paper are bounds on the oriented chromatic number of particular families of planar graphs, namely 2-dimensional grids, fat trees and fat fat trees.