The balanced decomposition number f(G) of a graph G was introduced by Fujita and Nakamigawa [Discr. Appl. Math., 156 (2008), pp. 3339-3344]. A balanced colouring of a graph G is a colouring of some of the vertices of G with two colours, such that there is the same number of vertices in each colour. Then, f(G) is the minimum integer s with the following property: For any balanced colouring of G,...

Let F be a function from Fpn to itself and δ a positive integer. F is called zerodifference δ-balanced if the equation F (x+a)−F (x) = 0 has exactly δ solutions for all nonzero a ∈ Fpn . As a particular case, all known quadratic planar functions are zero-difference 1-balanced; and some quadratic APN functions over F2n are zerodifference 2-balanced. In this paper, we study the relationship betwe...

Reiman’s inequality for the size of bipartite graphs of girth six is generalised to girth eight. It is optimal in as far as it admits the algebraic structure of generalised quadrangles as case of equality. This enables us to obtain the optimal estimate e ∼ v for balanced bipartite graphs. We also get an optimal estimate for very unbalanced graphs.

An extremely simple, linear time algorithm is given for constructing a domatic partition in totally balanced hypergraphs. This simpli es and generalizes previous algorithms for interval and strongly chordal graphs. On the other hand, the domatic number problem is shown to be NP-complete for several families of perfect graphs, including chordal and bipartite graphs.