There has been a recent explosion in the size of stored data, partially due to advances in storage technology, and partially due to the growing popularity of cloud-computing and the vast quantities of data generated. This motivates the need for streaming algorithms that can compute approximate solutions without full random access to all of the data. We model the problem of loading a graph onto ...

A graph is balanced if it is bipartite and every induced cycle has length divisible by four. In his book [6], Gérard Cornuéjols proposed a number of open questions, offering $5000 for the solution of any of them. Here we solve three of them, about balanced graphs.

In this paper we prove that any distance-balanced graph G with ∆(G) ≥ |V (G)| − 3 is regular. Also we define notion of distancebalanced closure of a graph and we find distance-balanced closures of trees T with ∆(T ) ≥ |V (T )| − 3.

Graph partitioning is a classical graph theory problem that has proven to be NP-hard. Most of the research in literature has focused its attention on a particular case of the problem called the graph bisection problem, where k = 2, such that the parts have approximately equal weight and minimizing the size of the edge cut. In this article, we describe how to obtain balanced partitioning on a gi...

In this paper, we show that each f -graph with more than 5 vertices has an unique minimum bipartition, and we also show the upper and lower bound of the balanced size of f -graphs.

A graph G is Berge if no induced subgraph of G is an odd cycle of length at least five or the complement of one; and it is known that this is equivalent to being perfect, that is, that the chromatic number of every induced subgraph equals the size of its largest clique. A skew partition in G is a partition (A,B) of V (G) such that G[A] is not connected and G[B] is not connected; and it is balan...

A colouring of a strongly regular graph is an allocation of colours (or treatments) to the vertices of the graph. Such a colouring is balanced if every pair of distinct colours occurs equally often on the ends of an edge. When the graph is the complete regular multipartite graph a balanced colouring is just a balanced incomplete-block design, or 2-design. Some constructions are given. For examp...

A graph G = (V,E) with p vertices and q edges is said to be a total mean cordial graph if there exists a function f : V (G) → {0, 1, 2} such that f(xy) = [(f(x)+f(y))/2] where x, y ∈ V (G), xy ∈ E(G), and the total number of 0, 1 and 2 are balanced. That is |evf (i) − evf (j)| ≤ 1, i, j ∈ {0, 1, 2} where evf (x) denotes the total number of vertices and edges labeled with x (x = 0, 1, 2). In thi...

We prove Ramsey-type results for intersection graphs of geometric objects the plane. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0 such that for every family F of n ≥ 2 convex sets in the plane, the intersection graph of F or its complement contains a balanced complete bipartite graph of size at least cn. There is a con...

A biased graph consists of a graph G together with a collection of distinguished cycles of G, called balanced, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs on G arise from orienting G and then labelling the edges of G with elements of a group Γ. In this case, we may define a biased graph by declaring a cycle to be balanced...