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-...

For a connected graph, a path containing all vertices is known as Hamiltonian path. For general graphs, there is no known necessary and sufficient condition for the existence of Hamiltonian paths and the complexity of finding a Hamiltonian path in general graphs is NP-Complete. We present a necessary and sufficient condition for the existence of Hamiltonian paths in 2-trees. Using our character...

Although there have been a lot of efforts to seek nice characterization of non-Hamiltonian graphs, little progress has been made so far. An important progress was achieved by Chvital [5, 61 who introduced the class of non-l-tough graphs (Nl T) and the class of non-sub-2-factor graphs (NS2F). Both contain only non-Hamiltonian graphs and the conditions for membership can be checked in non-determi...

Competition graphs were rst introduced by Joel Cohen in the study of food webs and have since been extensively studied. Graphs which are the competition graph of a strongly connected or Hamiltonian digraph are of particular interest in applications to communication networks. It has been previously established that every graph without isolated vertices (except K 2) which is the competitiongraph ...

Two new series of graphs are introduced. Properties of these graphs make them suitable as test beds for Hamiltonian path heuristics. The graphs are planar and regular of degree 3, and each series has a simple recursive definition. These graphs are “conceptual adversaries” to methods that try to exploit local characteristics. For each series, we give an analysis showing that the number of Hamilt...

A graphG is k-ordered if for any sequence of k distinct vertices ofG, there exists a cycle in G containing these k vertices in the specified order. It is k-ordered Hamiltonian if, in addition, the required cycle is Hamiltonian. The question of the existence of an infinite class of 3-regular 4-ordered Hamiltonian graphs was posed in 1997 [10]. At the time, the only known examples were K4 and K3,...

It is shown that every connected vertex-transitive graph of order 4p, where p is a prime, is hamiltonian with the exception of the Coxeter graph which is known to possess a Hamilton path. In 1969, Lovász [22] asked if every finite, connected vertex-transitive graph has a Hamilton path, that is, a path going through all vertices of the graph. With the exception of K 2 , only four connected verte...

One of the well-known NP-complete problems in graph theory is the Hamiltonian path problem; i.e., finding a simple path in the graph such that every vertex visits exactly once [5]. The two-dimensional integer grid G is an infinite undirected graph in which vertices are all points of the plane with integer coordinates and two vertices are connected by an edge if and only if the Euclidean distanc...

For the interconnection network topology, it is usually represented by a graph. When a network is used, processors and/or links faults may happen. Thus, it is meaningful to consider faulty networks. We consider k-regular graphs in this paper. We define a k-regular hamiltonian and hamiltonian connected graph G is super fault-tolerant hamiltonian if G remains hamiltonian after removing at most k ...

The presence of edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the network. Edge-disjoint Hamiltonian cycles also provide the edge-fault tolerant hamiltonicity of an interconnection network. In this paper, we first study the property of edge-disjoint Hamiltonian cycles in tr...