Hajo Broersma

Not every 2-tough graph is Hamiltonian

We present (9/4-e)-tough graphs without a Hamilton path for arbitrary >0, thereby refuting a well-known conjecture due to Chvatal. We also present (7/4-e) -tough chordal graphs without a Hamilton path for any e>0.

A graph covering algorithm for a coarse grain reconfigurable system

The availability of high-level design entry tooling is crucial for the viability of any reconfigurable SoC architecture. This paper presents a graph covering algorithm. The graph covering is done in two steps: template generation and template selection. The objective of template generation step is to extract functional equivalent structures, i.e. templates, from a control data flow graph. By inspecting the graph, the algorithm generates all the possible templates and the corresponding matches. Using unique serial numbers and circle numbers, the algorithm can find all distinct templates with multiple outputs. The template selection algorithm shows how this information can be used in compilers for reconfigurable systems. The objective of the template selection algorithm is to find an efficient cover for an application graph with a minimal number of distinct templates and minimal number of matches.

A linear time algorithm for minimum fill-in and treewidth for distance hereditary graphs

A graph is distance hereditary if it preserves distances in all its connected induced subgraphs. The MINIMUM FILL-IN problem is the problem of finding a chordal supergraph with the smallest possible number of edges. The TREEWIDTH problem is the problem of finding a chordal embedding of the graph with the smallest possible clique number. In this paper we show that both problems are solvable in linear time for distance hereditary graphs.

Cycles through subsets with large degree sums

Let G be a 2-connected graph on n vertices and let X V(G). We say that G is X-cyclable if G has an X-cycle, i.e., a cycle containing all vertices of X. We denote by ?(X) the maximum number of pairwise nonadjacent vertices in the subgraph G[X] of G induced by X. If G[X] is not complete, we denote by ?(X) the minimum cardinality of a set of vertices of G separating two vertices of X. By ?(X) we denote the minimum degree (in G) of the vertices of X, and by ?3(X) the minimum value of the degree sum (in G) of any three pairwise nonadjacent vertices of X. Our first main result is the following extension in terms of X-cyclability of a result on hamiltonian graphs by Bauer et al. If ?3(X) n + mingk(X), ?(X), then G is X-cyclable. Our second main result is the following generalization of a result of Fournier. If ?(X) ?(X), then G is X-cyclable. We give a number of extensions of other known results, thereby generalizing some recent results of Veldman.

A generalization of a result of Ha¨ggkvist and Nicoghossian

Using a variation of the Bondy-Chvatal closure theorem the following result is proved: If G is a 2-connected graph with n vertices and connectivity κ such that d(x) + d(y) + d(z) ≥ n + κ for any triple of independent vertices x, y, z, then G is hamiltonian.

Heavy cycles in weighted graphs

An (edge-)weighted graph is a graph in which each edge e is assigned a nonnegative real number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges, and an optimal cycle is one of maximum weight. The weighted degree w(v) of a vertex v is the sum of the weights of the edges incident with v. The following weighted analogue (and generalization) of a well-known result by Dirac for unweighted graphs is due to Bondy and Fan. Let G be a 2-connected weighted graph such that w(v) ≥ r for every vertex v of G. Then either G contains a cycle of weight at least 2r or every optimal cycle of G is a Hamilton cycle. We prove the following weighted analogue of a generalization of Diracs result that was first proved by Posa. Let G be a 2-connected weighted graph such that w(u)+w(v) ≥ s for every pair of nonadjacent vertices u and v. Then G contains either a cycle of weight at least s or a Hamilton cycle. Examples show that the second conclusion cannot be replaced by the stronger second conclusion from the result of Bondy and Fan. However, we characterize a natural class of edge-weightings for which these two conclusions are equivalent, and show that such edge-weightings can be recognized in time linear in the number of edges.

Closure concepts - a survey

Abstract. In this paper we survey results of the following type (known as closure results). Let P be a graph property, and let C(u,v) be a condition on two nonadjacent vertices u and v of a graph G. Then G+uv has property P if and only if G has property P. The first and now well-known result of this type was established by Bondy and Chvátal in a paper published in 1976: If u and v are two nonadjacent vertices with degree sum n in a graph G on n vertices, then G+uv is hamiltonian if and only if G is hamiltonian. Based on this result, they defined the n-closure cln (G) of a graph G on n vertices as the graph obtained from G by recursively joining pairs of nonadjacent vertices with degree sum n until no such pair remains. They showed that cln(G) is well-defined, and that G is hamiltonian if and only if cln(G) is hamiltonian. Moreover, they showed that cln(G) can be obtained by a polynomial algorithm, and that a Hamilton cycle in cln(G) can be transformed into a Hamilton cycle of G by a polynomial algorithm. As a consequence, for any graph G with cln(G)=Kn (and n≥3), a Hamilton cycle can be found in polynomial time, whereas this problem is NP-hard for general graphs. All classic sufficient degree conditions for hamiltonicity imply a complete n-closure, so the closure result yields a common generalization as well as an easy proof for these conditions. In their first paper on closures, Bondy and Chvátal gave similar closure results based on degree sum conditions for nonadjacent vertices for other graph properties. Inspired by their first results, many authors developed other closure concepts for a variety of graph properties, or used closure techniques as a tool for obtaining deeper sufficiency results with respect to these properties. Our aim is to survey this progress on closures made in the past (more than) twenty years.

Backbone Colorings for Networks

We study backbone colorings, a variation on classical vertex colorings: Given a graph G=(V,E) and a spanning subgraph H (the backbone) of G, a backbone coloring for G and H is a proper vertex coloring V →{ 1,2,... } in which the colors assigned to adjacent vertices in H differ by at least two. We concentrate on the cases where the backbone is either a spanning tree or a spanning path.

More about subcolorings

A subcoloring is a vertex coloring of a graph in which every color class induces a disjoint union of cliques. We derive a number of results on the combinatorics, the algorithmics, and the complexity of subcolorings.On the negative side, we prove that 2-subcoloring is NP-hard for comparability graphs, and that 3-subcoloring is NP-hard for AT-free graphs and for complements of planar graphs. On the positive side, we derive polynomial time algorithms for 2-subcoloring of complements of planar graphs, and for r-subcoloring of interval and of permutation graphs. Moreover, we prove asymptotically best possible upper bounds on the subchromatic number of interval graphs, chordal graphs, and permutation graphs in terms of the number of vertices.

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.