for a graph g, let d(g) be the set of all strong orientations of g. the orientationnumber of g is min {d(d) |d belongs to d(g)}, where d(d) denotes the diameterof the digraph d. in this paper, we determine the orientation number for some subgraphs of complete bipartite graphs.

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. In this paper we introduce a new graph associated to modules over commutative rings. We study the relationship between the algebraic properties of modules and their associated graphs. A topological characterization for the completeness of the special subgraphs is presented. Also modules whose associated graph is complete...

Given a graph and a positive integer k, the biclique vertex-partition problem asks whether the vertex set of the graph can be partitioned into at most k bicliques (connected complete bipartite subgraphs). It is known that this problem is NP-complete for bipartite graphs. In this paper we investigate the computational complexity of this problem in special subclasses of bipartite graphs. We prove...

In graphs of bounded arboricity, the total complexity of all maximal complete bipartite subgraphs is O(n). We describe a linear time algorithm to list such subgraphs. The arboricity bound is necessary: for any constant k and any n there exists an n-vertex graph with O(n) edges and (n/ log n) maximal complete bipartite subgraphs Kk,`. ∗Work supported in part by NSF grant CCR-9258355.

Let F be a finite field, G = (V, E) be an undirected graph on n vertices, and let S(F,G) be the set of all symmetric n× n matrices over F whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let mr(F,G) be the minimum rank of all matrices in S(F,G). If F is a finite field with pt elements, p = 2, it is shown that mr(F,G) ≤ 2 if and only if the compl...