Parallel O(log(n)) time edge-colouring of trees and Halin graphs

We present parallel O(log(n))-time algorithms for optimal edge colouring of trees and Halin graphs with n processors on a a parallel random access machine without write conflicts (P-RAM). In the case of Halin graphs with a maximum degree of three, the colouring algorithm automatically finds every Hamiltonian cycle of the graph

Heuristic crossing minimisation algorithms for the two-page drawing problem

The minimisation of edge crossings in a book drawing of a graph G is one of the important goals for a linear VLSI design, and the two-page crossing number of a graph G provides an upper bound for the standard planar crossing number. We propose several new heuristics for the two-page drawing problem, and test them on benchmark test suites, Rome graphs and Random Connected Graphs. We also test some typical graphs, and get some exact results. The results for some circulant graphs are better than the one presented by Cimikowski. We have a conjecture for cartesian graphs, supported by our experimental results, and provide direct methods to get optimal solutions for 3- or 4-row meshes and Halin graphs

One- and two-page crossing numbers for some types of graphs

One- and two-page crossing numbers for some types of graph

Various heuristic algorithms to minimise the two-page crossing numbers of graphs

We propose several new heuristics for the twopage book crossing problem, which are based on recent algorithms for the corresponding one-page problem. Especially, the neural network model for edge allocation is combined for the first time with various one-page algorithms. We investigate the performance of the new heuristics by testing them on various benchmark test suites. It is found out that the new heuristics outperform the previously known heuristics and produce good approximations of the planar crossing number for severalwell-known graph families. We conjecture that the optimal two-page drawing of a graph represents the planar drawing of the graph

Oriented coloring on recursively defined digraphs

Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G=(V,A) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows, that finding the chromatic number of an oriented graph is an NP-hard problem. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.Comment: 14 page