This volume collects the extended abstracts of the contributions that have been selected for presentation at the Workshop. As it was the case with previous workshops, we are planning to edit a special , fully refereed, proceedings volume with Discrete Applied Mathematics or Discrete Optimization and hereby invite all participants to submit full-length papers relating to the topics of the Worksh...

S of the Ghent Graph Theory Workshop on Longest Paths and Longest Cycles Kathie Cameron Degree-constrained spanning trees 2 Jan Goedgebeur Finding minimal obstructions to graph coloring 3 Jochen Harant On longest cycles in essentially 4-connected planar graphs 3 Frantǐsek Kardoš Barnette was right: not only fullerene graphs are Hamiltonian 4 Gyula Y. Katona Complexity questions for minimally t-...

A graph G is t-tough if any induced subgraph of it with x > 1 connected components is obtained from G by deleting at least tx vertices. It is shown that for every t and g there are t-tough graphs of girth strictly greater than g. This strengthens a recent result of Bauer, van den Heuvel and Schmeichel who proved the above for g = 3, and hence disproves in a strong sense a conjecture of Chvatal ...

Let G x H denote the Kronecker product of graphs G and H. Principal results are as follows: (a) If m is even and n 0 (mod 4), then one component of P,.+l x P,+1, and each component of each of CA x Pn+l, Pm+l x (7, and Cm x C, are edge decomposable into cycles of uniform length rs, where r and s are suitable divisors of m and n, respectively, (b) if m and n are both even, then each component of ...

Moon [J. Combin. Inform. System Sci. 19 (1994), 207–214] showed that every strong tournament contains a Hamiltonian cycle through at least three pancyclic arcs. In this paper, we extend the result of Moon and prove that if D is a strong c-partite tournament with c ≥ 3, then D contains a cycle C containing vertices from exactly c partite sets such that C contains at least three arcs, each of whi...

The locally twisted cube LTQn which is a newly introduced interconnection network for parallel computing is a variant of the hypercube Qn . Yang et al. [X. Yang, G.M. Megson, D.J. Evans, Locally twisted cubes are 4-pancyclic, Applied Mathematics Letters 17 (2004) 919–925] proved that LTQn is Hamiltonian connected and contains a cycle of length from 4 to 2 n for n ≥ 3. In this work, we improve t...

The crossed cube CQn introduced by Efe has many properties similar to those of the popular hypercube. However, the diameter of CQn is about one half of that of the hypercube. Failures of links and nodes in an interconnection network are inevitable. Hence, in this paper, we consider the hybrid fault-tolerant capability of the crossed cube. Letting fe and fv be the numbers of faulty edges and ver...

The directed anti-Oberwolfach problem asks for a 2-factorization (each factor has in-degree 1 and out-degree 1 for a total degree of two) of K2n+1, not with consistent cycle components in each 2-factor like the Oberwolfach problem, but such that every admissible cycle size appears at least once in some 2-factor. The solution takes advantage of both Piotrowski’s decomposition techniques used to ...

In this short note I discuss two problems which we discussed during the hyper-graph meeting. I. The following conjecture is attributed to I. Zarins by Bondy and Chvatal : Let k be an integer and n > n 0 (k). Then if G(n) (G(n) is a graph of n ver-tices) is Hamiltonian and does not have k pairwise independent vertices (i .e. the complementary graph does not contain a complete subgraph of k verti...