Matthias Kriesell

On decomposing a hypergraph into k connected sub-hypergraphs

By applying the matroid partition theorem of J. Edmonds (J. Res. Nat. Bur. Standards Sect. B 69 (1965) 67) to a hypergraphic generalization of graphic matroids, due to Lorea (Cahiers Centre Etudes Rech. Oper. 17 (1975) 289), we obtain a generalization of Tuttes disjoint trees theorem for hypergraphs. As a corollary, we prove for positive integers k and q that every (kq)-edge-connected hypergraph of rank q can be decomposed into k connected sub-hypergraphs, a well-known result for q = 2. Another by-product is a connectivity-type sufficient condition for the existence of k edge-disjoint Steiner trees in a bipartite graph.

Edge-disjoint trees containing some given vertices in a graph

We show that for any two natural numbers k, l there exist (smallest natural numbers fl(k)(gl(k)) such that for any fl(k)-edge-connected (gl(k)-edge-connected) vertex set A of a graph G with |A| ≤ l(|V(G) - A| ≤ l) there exists a system J of k edge-disjoint trees such that A ⊆ V(T) for each T ∈ J. We determine f3(k) = ⌊ 8k+3/6 ⌋. Furthermore, we determine for all natural numbers l,k the smallest number fl*(k) such that every fl*(k)-edge-connected graph on at most l vertices contains a system of k edge-disjoint spanning trees, and give applications to line graphs.

A Survey on Contractible Edges in Graphs of a Prescribed Vertex Connectivity

Abstract. The aim of the present paper is to survey old and recent results on contractible edges in graphs of a given vertex connectivity.

Cayley DHTs — a group-theoretic framework for analyzing DHTs based on cayley graphs

Static DHT topologies influence important features of such DHTs such as scalability, communication load balancing, routing efficiency and fault tolerance. While obviously dynamic DHT algorithms which have to approximate these topologies for dynamically changing sets of peers play a very important role for DHT networks, important insights can be gained by clearly focussing on the static DHT topology as well. In this paper we analyze and classify current DHTs in terms of their static topologies based on the Cayley graph group-theoretic model and show that most DHT proposals use Cayley graphs as static DHT topologies, thus taking advantage of several important Cayley graph properties such as vertex/edge symmetry, decomposability and optimal fault tolerance. Using these insights, Cayley DHT design can directly leverage algebraic design methods to generate high-performance DHTs adopting Cayley graph based static DHT topologies, extended with suitable dynamic DHT algorithms.

Asymptotically optimal K k -packings of dense graphs via fractional K k -decompositions

Let H be a fixed graph. A fractional H-decomposition of a graph G is an assignment of nonnegative real weights to the copies of H in G such that for each e ∈ E(G), the sum of the weights of copies of H containing e in precisely one. An H-packing of a graph G is a set of edge disjoint copies of H in G. The following results are proved. For every fixed k > 2, every graph with n vertices and minimum degree at least n(1 − 1/9k) + o(n) has a fractional Kk-decomposition and has a Kk-packing which covers all but o(n ) edges.

Contractible Subgraphs in 3-Connected Graphs

A subgraph H of a 3-connected finite graph G is called contractible if H is connected and G?V(H) is 2-connected. This work is concerned with a conjecture of McCuaig and Ota which states that for any given k there exists an f(k) such that any 3-connected graph on at least f(k) vertices possesses a contractible subgraph on k vertices. We prove this for k?4 and consider restrictions to maximal planar graphs, Halin graphs, line graphs of 6-edge-connected graphs, 5-connected graphs of bounded degree, and AT-free graphs.

A Degree Sum Condition for the Existence of a Contractible Edge in a κ-Connected Graph

It is known that a noncomplete ?-connected graph of minimum degree of at least ?5?4? contains a ?-contractible edge, i.e., an edge whose contraction yields again a ?-connected graph. Here we prove the stronger statement that a noncomplete ?-connected graph for which the sum of the degrees of any two distinct vertices is at least 2?54???1 possesses a ?-contractible edge. The bound is sharp and remains valid and sharp if we look only at degree sums at pairs of vertices at distances of one or two, provided that ??7.

Global Connectivity And Expansion: Long Cycles and Factors In f -Connected Graphs

Given a function f : ℕ→ℝ, call an n-vertex graph f-connected if separating off k vertices requires the deletion of at least f(k) vertices whenever k≤(n−f(k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f-connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-factor and a 2-factor if f(k)≥2k+1, and contains a Hamilton cycle if f(k)≥2(k+1)2. We conjecture that linear growth of f suffices to imply hamiltonicity.

All 4-connected Line Graphs of Claw Free Graphs Are Hamiltonian Connected

Thomassen conjectured that every 4-connected line graph is hamiltonian. Here we shall see that 4-connected line graphs of claw free graphs are hamiltonian connected.

On the Effect of Group Structures on Ranking Strategies in Folksonomies

Folksonomies have shown interesting potential for improving information discovery and exploration. Recent folksonomy systems explore the use of tag assignments, which combine Web resources with annotations (tags), and the users that have created the annotations. This article investigates on the effect of grouping resources in folksonomies, i.e. creating sets of resources, and using this additional structure for the tasks of search & ranking, and for tag recommendations. We propose several group-sensitive extensions of graph-based search and recommendation algorithms, and compare them with non group-sensitive versions. Our experiments show that the quality of search result ranking can be significantly improved by introducing and exploiting the grouping of resources (one-tailed t-Test, level of significance α=0.05). Furthermore, tag recommendations profit from the group context, and it is possible to make very good recommendations even for untagged resources– which currently known tag recommendation algorithms cannot fulfill.