We prove that there is a Steiner triple system T such that every simple cubic graph can have its edges coloured by points of T in such a way that for each vertex the colours of the three incident edges form a triple in T . This result complements the result of Holroyd and Škoviera that every bridgeless cubic graph admits a similar colouring by any Steiner triple system of order greater than 3. ...

edge distance-balanced graphs are graphs in which for every edge $e = uv$ the number of edges closer to vertex $u$ than to vertex $v$ is equal to the number of edges closer to $v$ than to $u$. in this paper, we study this property under some graph operations.

In $1994,$ degree distance  of a graph was introduced by Dobrynin, Kochetova and Gutman. And Gutman proposed the Gutman index of a graph in $1994.$ In this paper, we introduce the concepts of  multiplicative version of degree distance and the multiplicative version of Gutman index of a graph. We find the sharp upper bound for the  multiplicative version of degree distance and multiplicative ver...

Selected-internal Steiner minimum tree problem is a generalization of original Steiner minimum tree problem. Given a weighted complete graph G = (V,E) with weight function c, and two subsets R ′ ( R ⊆ V with |R − R ′ | ≥ 2, selected-internal Steiner minimum tree problem is to find a Steiner minimum tree T of G spanning R such that any leaf of T does not belong to R ′ . In this paper, suppose c ...

Let R be a finite set of terminals in a metric space (M,d). We consider finding a minimum size set S ⊆ M of additional points such that the unit-disc graph G[R ∪ S] of R ∪ S satisfies some connectivity properties. In the Steiner Tree with Minimum Number of Steiner Points (ST-MSP) problem G[R ∪ S] should be connected. In the more general Steiner Forest with Minimum Number of Steiner Points (SF-M...

Given a graph G = (V, E) with a length function on edges and a subset R of V , the full Steiner tree is defined to be a Steiner tree in G with all the vertices of R as its leaves. Then the full Steiner tree problem is to find a full Steiner tree in G with minimum length, and the bottleneck full Steiner tree problem is to find a full Steiner tree T in G such that the length of the largest edge i...

The Group Steiner Problem (GSP) is a generalized version of the well known Steiner Problem. For an undirected, connected distance graph with groups of required vertices and Steiner vertices, GSP asks for a shortest connected subgraph, containing at least one vertex of each group. As the Steiner Problem is NP-hard, GSP is too, and we are interested in approximation algorithms. EEcient approximat...

A new Genetic Algorithm (GA) for the Steiner Problem in a Graph (SPG) is presented. The algorithm is based on a bit-string encoding. A bitstring specifies selected Steiner vertices and the corresponding Steiner tree is computed using the Distance Network Heuristic. This scheme ensures that every bitstring correspond to a valid Steiner tree and thus eliminate the need for penalty terms in the co...