Let S be a Steiner triple system and G a cubic graph. We say that G is S-colourable if its edges can be coloured so that at each vertex the incident colours form a triple of S. We show that if S is a projective system PG(n, 2), n ≥ 2, then G is S-colourable if and only if it is bridgeless, and that every bridgeless cubic graph has an S-colouring for every Steiner triple system of order greater ...

Given a connected graph G = (V,E) (undirected, without loops and multiple edges) with positive edge costs (called also lengths) and a set Z ⊂ V of special (distinguished) vertices, the Steiner problem on graphs (networks) asks for a minimum cost tree within G that spans all members of Z. If |Z| = 2 we have the shortest path problem and if Z = V we get the minimum spanning tree problem, which ar...

We prove that a partial Steiner triple system 8 of order n can be embedded in a Steiner triple system T of any given admissible order greater than 4w. Furthermore, if G(S), the missing-edge graph of S, has the property that A(G)<ri(n + l)l and \E(G)\ then # can be embedded in a Steiner triple system of order 2n +1, provided that 2w +1 is admissible. We also prove that if there is a partial Stei...

An extended 1-perfect trade is a pair (T0, T1) of two disjoint binary distance-4 even-weight codes such that the set of words at distance 1 from T0 coincides with the set of words at distance 1 from T1. Such trade is called primary if any pair of proper subsets of T0 and T1 is not a trade. Using a computer-aided approach, we classify nonequivalent primary extended 1-perfect trades of length 10,...

The node-weighted Steiner tree problem is a variation of classical Steiner minimum tree problem. Given a graph G = (V,E) with node weight function C : V → R and a subset X of V , the node-weighted Steiner tree problem is to find a Steiner tree for the set X such that its total weight is minimum. In this paper, we study this problem in unit disk graphs and present a (1+ε)-approximation algorithm...

Given a point set K of terminals in the plane, the octilinear Steiner tree problem is to find a shortest tree that interconnects all terminals and edges run either in horizontal, vertical, or ±45 diagonal direction. This problem is fundamental for the novel octilinear routing paradigm in VLSI design, the socalled X-architecture. As the related rectilinear and the Euclidian Steiner tree problem ...

We study the problem of finding minimum multicuts for an undirected planar graph, where all the terminal vertices are on the boundary of the outer face. This is known as an Okamura-Seymour instance. We show that for such an instance, the minimum multicut problem can be reduced to the minimum-cost Steiner forest problem on a suitably defined dual graph. The minimum-cost Steiner forest problem ha...

In this paper, a parallel algorithm for the Steiner tree problem is presented. The algorithm is based on the well-known multi-start paradigm the GRASP and the well-known proximity structures from computational geometry. The main contribution of this paper is the O(n log n+log n log( n log n )) parallel algorithm for computing Steiner tree on the Euclidean plane. The parallel algorithm used prox...

We study global routing of multiterminal nets. We propose a new global router: each step consists of finding a tree, called a Steiner min-max tree, that is a Steiner tree with maximum-weight edge minimized (real vertices represent channels containing terminals of a net, Steiner vertices represent intermediate channels, and weights correspond to densities). We propose an 0 (min { e loglog e , n*...