Popular social networking applications such as Facebook, Twitter, Friendster, etc. generate very large graphs with different characteristics. These social networks are huge, comprising millions of nodes and edges that push existing graph mining algorithms and architectures to their limits. In product-rating graphs, users connect with each other and rate items in tandem. In such bipartite graphs...

Polynomial algorithms are given for the following two problems: • given a graph with n vertices and m edges, where m ≥ 3n3/2, find a complete balanced bipartite subgraph with parts about lnn ln(n/m) , • given a graph with n vertices, find a decomposition of its edges into complete balanced bipartite graphs having altogether O(n2/ lnn) vertices. Previous proofs of the existence of such objects, ...

We define a new kind of crossing number which generalizes both the bipartite crossing number and the outerplanar crossing number. We calculate exact values of this crossing number for many complete bipartite graphs and also give a lower bound. 1 Preliminaries The bipartite crossing number of a bipartite graph G was defined by Watkins in [W] to be the minimum number of crossings over all biparti...

It is known that if a bipartite graph G with n edges possesses any of three types of ordered labelings, then the complete graphK2nx+1 admits a cyclic G-decomposition for every positive integer x. We introduce variations of the ordered labelings and show that whenever a bipartite graph G admits one of these labelings, then there exists a cyclic G-decomposition of an infinite family of circulant ...

Kreweras’ conjecture [1] asserts that any perfect matching of the hypercubeQd, d ≥ 2, can be extended to a Hamiltonian cycle. We proved this conjec-ture in [2] but here we present an idea of a simplified proof. The matchinggraph M(G) of a graph G is a graph whose vertex set consists of all per-fect matchings of G, with two vertices being adjacent whenever the unionof the cor...

In the edge precoloring extension problem we are given a graph with some of the edges having a preassigned color and it has to be decided whether this coloring can be extended to a proper k-edge-coloring of the graph. In list edge coloring every edge has a list of admissible colors, and the question is whether there is a proper edge coloring where every edge receives a color from its list. We s...

Let G be a subgraph of a complete bipartite graph Kn,n. Let N(G) be a 0-1 incidence matrix with edges of Kn,n against images of G under the automorphism group of Kn,n. A diagonal form of N(G) is found for every G, and whether the row space of N(G) over Zp contains the vector of all 1’s is determined. This re-proves Caro and Yuster’s results on zero-sum bipartite Ramsey numbers [3], and provides...

The binding number of a graph was introduced by D.R.Woodall in 1973 [10] and is defined as the minimum of the ratios | (X)| /|X| taken over all non-empty subsets ofX of vertices inG such that (X) = V (G), where (X) = ∪v∈X (v) and (v) the set of all vertices adjacent to a vertex v in G. We obtain exact values of the binding number of middle graph of cycle, path, complete graph and complete bipar...