Thomassen conjectured [8] that every 4-connected line graph is hamiltonian. An hourglass is a graph isomorphic to K5−E(C4), where C4 is a cycle of length 4 in K5. In [2], it is shown that every 4-connected line graph without an induced subgraph isomorphic to the hourglass is hamiltonian connected. In this note, we prove that every 3-connected, essentially 4-connected hourglass free line graph i...

This project will contribute to the departmental research theme Critical Systems in that it develops a method for generating (black-box) test data with a uniform distribution guarantee for programs working on graph-like structures. This includes test data for classical graph algorithms but also, for example, pointer-data structures for testing C programs. The uniform distribution ensures that t...

Let G be a claw-free graph and let cl(G) be the closure of G. We present a method for characterizing classes G i , i = 3; : : : ; 7, of 2-connected closed claw-free graphs with the following properties.

We prove that a 4-connected K 4;4-minor free graph on n vertices has at most 4n ? 8 edges and we use this result to show that every K 4;4-minor free graph has vertex-arboricity at most 4. This improves the case (n; m) = (7; 3) of the following conjecture of Woodall: the vertexset of a graph without a K n-minor and without a K b n+1 2 c;d n+1 2 e-minor can be partitioned in n ? m + 1 subgraphs w...

This paper proposes an original approach to cluster multi-component data sets, including an estimation of the number of clusters. From the construction of a minimal spanning tree with Prim’s algorithm, and the assumption that the vertices are approximately distributed according to a Poisson distribution, the number of clusters is estimated by thresholding the Prim’s trajectory. The correspondin...

Let (G) denote the minimum number of edges to be removed from a graph G to make it bipartite. For each 3-chromatic graph F we determine a parameter (F ) such that for each F-free graph G on n vertices with minimum degree (G) 2n/( (F )+ 2)+ o(n) we have (G)= o(n2), while there are F-free graphs H with (H) ≥ 2n/( (F )+ 2) for which (H)= (n2). © 2007 Elsevier B.V. All rights reserved. MSC: primary...

Every triangle-free planar graph on n vertices has an independent set of size at least (n+ 1)/3, and this lower bound is tight. We give an algorithm that, given a triangle-free planar graph G on n vertices and an integer k ≥ 0, decides whether G has an independent set of size at least (n+ k)/3, in time 2 √ n. Thus, the problem is fixed-parameter tractable when parameterized by k. Furthermore, a...

Three results on the interval number of a graph on n vertices are presented. (1) The interval number of almost every graph is between n/4lg n and n/4 (this also holds for almost every bipartite graph). (2) There exist Km,+ -free bipartite graphs with interval number at least c(m)n’-*I( , m+lj/lg n which can be improved to &$4+0(&) for m = 2 and (n/2):/lg n for m = 3. (3) There exists a regular ...

It is known that ff a 2-connected graph G of suiBciently large order n satisfies the property that the union of the neighborhoods of each pair of vertices has cardinallty at least ~-, " then G is hamiltonian. In this paper, we obtain a similar generalization of Dirac's Theorem for K(1, 3)-free graphs. In particular, we show that if G is a 2-connected K(1, 3)-free graph of order n with the cardi...