Definition (Maximal size complete bipartite induced subgraph). β(G) := size of maximal complete bipartite induced subgraph of G. Definition (Minimal bipartite decomposion number). τ(G) := minimal number of complete edge disjoint covering bipartite subgraphs of G. Definition (Minimal nontrivial bipartite decomposion number). τ (G) := minimal number of complete edge disjoint covering nontrivial (...

Given a tree and a set of paths in the tree, the problem of finding a minimum number of paths from the given path set to cover all the vertices in the tree is investigated in the paper. To distinguish from the classical path cover problem, such an optimization problem is referred to as vertex covering by paths. The problem and its edge variant, edge covering by paths, find applications in machi...

In this paper, a new algorithm for solving the single loop routing problem is presented. The purpose of the single loop routing problem(SLRP) is to find the shortest loop for an automated guided vehicle covering at least one edge of each department of a block layout. First it shown that this problem can be represented as a graph model. Then a meta-heuristic algorithm based on and colony system ...

The edge set of Kn cannot be decomposed into edge-disjoint hexagons (or 6-cycles) when n =1= 1 or 9 (mod 12). We discuss adding edges to the edge set of Kn so that the resulting graph can be decomposed into edge-disjoint hexagons. This paper gives the solution to this minimum covering of Kn with hexagons problem.

This paper discusses the graph covering problem in which a set of edges in an edgeand nodeweighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a natural linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP ...

In this paper, a new algorithm for solving the single loop routing problem is presented. The purpose of the single loop routing problem(SLRP) is to find the shortest loop for an automated guided vehicle covering at least one edge of each department of a block layout. First it shown that this problem can be represented as a graph model. Then a meta-heuristic algorithm based on and colony system ...

We characterize the edge-signed graphs in which every 'significant' positive closed walk (or combination of walks) has even length, under seven different criteria for significance, and also those edge-signed graphs whose double covering graph is bipartite. If the property of even length is generalized to positivity in a second edge signing, the characterizations generalize as well. We also char...