Optimal Groomings with Grooming Ratios Six and Seven

Grooming uniform all‐to‐all traffic in optical (SONET) rings with grooming ratio C requires the determination of a decomposition of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The determination of optimal C‐groomings has been considered for , and completely solved for . For , it has been shown that the lower bound for the drop cost of an optimal C‐grooming can be attained for almost all orders with 5 exceptions and 308 possible exceptions. For , there are infinitely many unsettled orders; especially the case is far from complete. In this paper, we show that the lower bound for the drop cost of a 6‐grooming can be attained for almost all orders by reducing the 308 possible exceptions to 3, and that the lower bound for the drop cost of a 7‐grooming can be attained for almost all orders with seven exceptions and 16 possible exceptions. Moreover, for the unsettled orders, we give upper bounds for the minimum drop costs.     

Optical grooming with grooming ratio eight

Grooming uniform all-to-all traffic in optical ring networks with grooming ratio C requires the determination of graph decompositions of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The minimum possible drop cost is determined for grooming ratio 8, and this cost is shown to be realized with six exceptions, and 37 possible exceptions, the largest being 105.     

Optical grooming with grooming ratio nine

Grooming uniform all-to-all traffic in optical ring networks with grooming ratio C requires the determination of graph decompositions of the complete graph into subgraphs each having at most C edges. The drop cost of such a grooming is the total number of vertices of nonzero degree in these subgraphs, and the grooming is optimal when the drop cost is minimum. The minimum drop cost is determined for grooming ratio 9. Previously this bound was shown to be met when with two exceptions and eleven additional possible exceptions for n, and also when with one exception and one possible exception for n. In this paper it is shown that the bound is met for all with four exceptions for n∈{8,11,14,17} and one possible exception for n=20. Using this result, it is further shown that when and n is sufficiently large, the bound is also met.     

A survey on the existence of G‐Designs

A G‐design of order n is a decomposition of the complete graph on n vertices into edge‐disjoint subgraphs isomorphic to G. We survey the current state of knowledge on the existence problem for G‐designs. This includes references to all the necessary designs and constructions, as well as a few new designs. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 373–410, 2008     

Global cycle properties in graphs with large minimum clustering coefficient

Let 𝒫 be a graph property. A graph G is said to be locally 𝒫 (closed locally 𝒫) if the subgraph induced by the open neighbourhood (closed neighbourhood, respectively) of every vertex in G has property 𝒫. The clustering coefficient of a vertex is the proportion of pairs of its neighbours that are themselves neighbours. The minimum clustering coefficient of G is the smallest clustering coefficient among all vertices of G. Let H be a subgraph of a graph G and let S ? V (H). We say that H is a strongly induced subgraph of G with attachment set S, if H is an induced subgraph of G and the vertices of V (H) ? S are not incident with edges that are not in H. A graph G is fully cycle extendable if every vertex of G lies in a triangle and for every nonhamiltonian cycle C of G, there is a cycle of length |V (C)|?+?1 that contains the vertices of C. A complete characterization, of those locally connected graphs with minimum clustering coefficient 1/2 and maximum degree at most 6 that are fully cycle extendable, is given in terms of forbidden strongly induced subgraphs (with specified attachment sets). Moreover, it is shown that all locally connected graphs with Δ?≤?6 and sufficiently large minimum clustering coefficient are weakly pancylic, thereby proving Ryj´ǎcek’s conjecture for this class of graphs.     

The inverse inertia problem for graphs: Cut vertices, trees, and a counterexample

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, which settles an open problem from the October 2006 AIM Workshop on Spectra of Families of Matrices described by Graphs, Digraphs and Sign Patterns. We also determine some restrictions on the inertia set of any graph.     

Steiner centers in graphs

The Steiner distance of a set S of vertices in a connected graph G is the minimum size among all connected subgraphs of G containing S. For n ≥ 2, the n-eccentricity e_n(ν) of a vertex ν of a graph G is the maximum Steiner distance among all sets S of n vertices of G that contains ν. The n-diameter of G is the maximum n-eccentricity among the vertices of G while the n-radius of G is the minimum n-eccentricity. The n-center of G is the subgraph induced by those vertices of G having minimum n-eccentricity. It is shown that every graph is the n-center of some graph. Several results on the n-center of a tree are established. In particular, it is shown that the n-center of a tree is a tree and those trees that are n-centers of trees are characterized.     

Complete partitions of graphs

A complete partition of a graph G is a partition of its vertex set in which any two distinct classes are connected by an edge. Let cp(G) denote the maximum number of classes in a complete partition of G. This measure was defined in 1969 by Gupta [19], and is known to be NP-hard to compute for several classes of graphs. We obtain essentially tight lower and upper bounds on the approximability of this problem. We show that there is a randomized polynomial-time algorithm that given a graph G with n vertices, produces a complete partition of size Ω(cp(G)/√lgn). This algorithm can be derandomized. We show that the upper bound is essentially tight: there is a constant C > 1, such that if there is a randomized polynomial-time algorithm that for all large n, when given a graph G with n vertices produces a complete partition into at least C·cp(G)/√lgn classes, then NP ⊆ RTime(n ^{O(lg lg n)}). The problem of finding a complete partition of a graph is thus the first natural problem whose approximation threshold has been determined to be of the form Θ((lgn) ^c ) for some constant c strictly between 0 and 1. The work reported here is a merger of the results reported in [30] and [21].     

Random Graph Coverings I: General Theory and Graph Connectivity

In this paper we describe a simple model for random graphs that have an n-fold covering map onto a fixed finite base graph. Roughly, given a base graph G and an integer n, we form a random graph by replacing each vertex of G by a set of n vertices, and joining these sets by random matchings whenever the corresponding vertices are adjacent in G. The resulting graph covers the original graph in the sense that the two are locally isomorphic. We suggest possible applications of the model, such as constructing graphs with extremal properties in a more controlled fashion than offered by the standard random models, and also "randomizing" given graphs. The main specific result that we prove here (Theorem 1) is that if is the smallest vertex degree in G, then almost all n-covers of G are -connected. In subsequent papers we will address other graph properties, such as girth, expansion and chromatic number. Received June 21, 1999/Revised November 16, 2000 RID="*" ID="*" Work supported in part by grants from the Israel Academy of Aciences and the Binational Israel-US Science Foundation.     

Partitioning the Vertices of a Graph into Two Total Dominating Sets

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study graphs whose vertex set can be partitioned into two total dominating sets. In particular, we develop several sufficient conditions for a graph to have a vertex partition into two total dominating sets. We also show that with the exception of the cycle on five vertices, every selfcomplementary graph with minimum degree at least two has such a partition.     

Density Conditions For Triangles In Multipartite Graphs

We consider the problem of finding a large or dense triangle-free subgraph in a given graph G. In response to a question of P. Erdős, we prove that, if the minimum degree of G is at least 9|V (G)|/10, the largest triangle-free subgraphs are precisely the largest bipartite subgraphs in G. We investigate in particular the case where G is a complete multipartite graph. We prove that a finite tripartite graph with all edge densities greater than the golden ratio has a triangle and that this bound is best possible. Also we show that an infinite-partite graph with finite parts has a triangle, provided that the edge density between any two parts is greater than 1/2.     

The rainbow connection of a graph is (at most) reciprocal to its minimum degree

An edge‐colored graph Gis rainbow edge‐connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make Grainbow edge‐connected. We prove that if Ghas nvertices and minimum degree δ then rc(G)<20n/δ. This solves open problems from Y. Caro, A. Lev, Y. Roditty, Z. Tuza, and R. Yuster (Electron J Combin 15 (2008), #R57) and S. Chakrborty, E. Fischer, A. Matsliah, and R. Yuster (Hardness and algorithms for rainbow connectivity, Freiburg (2009), pp. 243–254). A vertex‐colored graph Gis rainbow vertex‐connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex‐connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make Grainbow vertex‐connected. One cannot upper‐bound one of these parameters in terms of the other. Nevertheless, we prove that if Ghas nvertices and minimum degree δ then rvc(G)<11n/δ. We note that the proof in this case is different from the proof for the edge‐colored case, and we cannot deduce one from the other. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 185–191, 2010     

Sufficient conditions for graphs to be λ′‐optimal,super‐edge‐connected,and maximally edge‐connected

The restricted‐edge‐connectivity of a graph G, denoted by λ′(G), is defined as the minimum cardinality over all edge‐cuts S of G, where G‐S contains no isolated vertices. The graph G is called λ′‐optimal, if λ′(G) = ξ(G), where ξ(G) is the minimum edge‐degree in G. A graph is super‐edge‐connected, if every minimum edge‐cut consists of edges adjacent to a vertex of minimum degree. In this paper, we present sufficient conditions for arbitrary, triangle‐free, and bipartite graphs to be λ′‐optimal, as well as conditions depending on the clique number. These conditions imply super‐edge‐connectivity, if δ (G) ≥ 3, and the equality of edge‐connectivity and minimum degree. Different examples will show that these conditions are best possible and independent of other results in this area. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 228–246, 2005     

The reconstruction conjecture and edge ideals

Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebraic properties of the edge ideal from the deck of G, that is, from the collection of subgraphs obtained by removing a vertex from G. These properties include the Krull dimension, the Hilbert function, and all the graded Betti numbers β_i,j where j<n. We also state many further questions that arise from our study.     

Bounds on a generalized domination parameter

Abstract

If n is an integer, n ≥ 2 and u and v are vertices of a graph G, then u and v are said to be K_n-adjacent vertices of G if there is a subgraph of G, isomorphic to K_n , containing u and v. For n ≥ 2, a K_n- dominating set of G is a set D of vertices such that every vertex of G belongs to D or is K_n-adjacent to a vertex of D. The K_n-domination number γK_n (G) of G is the minimum cardinality among the K_n-dominating sets of vertices of G. It is shown that, for n ε {3,4}, if G is a graph of order p with no K_n-isolated vertex, then γK_n (G) ≤ p/n. We establish that this is a best possible upper bound. It is shown that the result is not true for n ≥ 5.     

List Coloring of Random and Pseudo-Random Graphs

choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). It is shown that the choice number of the random graph G(n, p(n)) is almost surely whenever . A related result for pseudo-random graphs is proved as well. By a special case of this result, the choice number (as well as the chromatic number) of any graph on n vertices with minimum degree at least in which no two distinct vertices have more than common neighbors is at most . Received: October 13, 1997     

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.     

Total domination in 2‐connected graphs and in graphs with no induced 6‐cycles

A set S of vertices in a graph G is a total dominating set of G if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number γ_t(G) of G. It is known [J Graph Theory 35 (2000), 21–45] that if G is a connected graph of order n > 10 with minimum degree at least 2, then γ_t(G) ≤ 4n/7 and the (infinite family of) graphs of large order that achieve equality in this bound are characterized. In this article, we improve this upper bound of 4n/7 for 2‐connected graphs, as well as for connected graphs with no induced 6‐cycle. We prove that if G is a 2‐connected graph of order n > 18, then γ_t(G) ≤ 6n/11. Our proof is an interplay between graph theory and transversals in hypergraphs. We also prove that if G is a connected graph of order n > 18 with minimum degree at least 2 and no induced 6‐cycle, then γ_t(G) ≤ 6n/11. Both bounds are shown to be sharp. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 55–79, 2009     

Forcing highly connected subgraphs

A theorem of Mader states that highly connected subgraphs can be forced in finite graphs by assuming a high minimum degree. We extend this result to infinite graphs. Here, it is necessary to require not only high degree for the vertices but also high vertex‐degree (or multiplicity) for the ends of the graph, that is, a large number of disjoint rays in each end. We give a lower bound on the degree of vertices and the vertex‐degree of the ends which is quadratic in k, the connectedness of the desired subgraph. In fact, this is not far from best possible: we exhibit a family of graphs with a degree of order 2^k at the vertices and a vertex‐degree of order k log k at the ends which have no k‐connected subgraphs. Furthermore, if in addition to the high degrees at the vertices, we only require high edge‐degree for the ends (which is defined as the maximum number of edge‐disjoint rays in an end), Mader's theorem does not extend to infinite graphs, not even to locally finite ones. We give a counterexample in this respect. But, assuming a lower bound of at least 2k for the edge‐degree at the ends and the degree at the vertices does suffice to ensure the existence (k + 1)‐edge‐connected subgraphs in arbitrary graphs. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 331–349, 2007     

On the maximum cardinality search lower bound for treewidth

The Maximum Cardinality Search (MCS) algorithm visits the vertices of a graph in some order, such that at each step, an unvisited vertex that has the largest number of visited neighbours becomes visited. A maximum cardinality search ordering (MCS-ordering) of a graph is an ordering of the vertices that can be generated by the MCS algorithm. The visited degree of a vertex v in an MCS-ordering is the number of neighbours of v that are before v in the ordering. The visited degree of an MCS-ordering ψ of G is the maximum visited degree over all vertices v in ψ. The maximum visited degree over all MCS-orderings of graph G is called its maximum visited degree. Lucena [A new lower bound for tree-width using maximum cardinality search, SIAM J. Discrete Math. 16 (2003) 345-353] showed that the treewidth of a graph G is at least its maximum visited degree.We show that the maximum visited degree is of size O(logn) for planar graphs, and give examples of planar graphs G with maximum visited degree k with O(k!) vertices, for all k∈N. Given a graph G, it is NP-complete to determine if its maximum visited degree is at least k, for any fixed k?7. Also, this problem does not have a polynomial time approximation algorithm with constant ratio, unless P=NP. Variants of the problem are also shown to be NP-complete.In this paper, we also propose some heuristics for the problem, and report on an experimental analysis of them. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the value of the maximum visited degree of a graph, which appear to give results close to optimal on many graphs from real life applications.