We consider special types of mod-λ flows, called odd and even mod-λ flows, for directed graphs, and prove that the numbers of such flows can be interpolated by polynomials in λ with the degree given by the cycle rank of the graph. The proofs involve computation of the number of integer solutions in a polyhedral region of Euclidean space using theorems due to Ehrhart. The resulting reciprocity properties of the interpolating polynomials for even flows are considered. The analogous properties of odd and even mod-λ potential differences and their associated potentials are also developed.

There is a close relationship between biased graph games and random graph processes. In this paper, we develop the analogy and give further interesting instances.

We consider the oriented binary tree and the oriented hypercube. The tree edges are oriented from the root to the leaves, while the orientation of the cube edges is induced by the direction from 0 to 1 in the coordinatewise form. The problem is to embed such a tree with l levels into the oriented n-cube as an oriented subgraph, for minimal possible n. A new approach to such problems is presented, which improves the known upper bound n/l ≤ 3/2 given by Havel [1] to n/l ≤ 4/3 + o(1) as l → ∞.

We prove that almost every r-regular digraph is Hamiltonian for all fixed r ≥ 3.

An infinite graph is called bounded if for every labelling of its vertices with natural numbers there exists a sequence of natural numbers which eventually exceeds the labelling along any ray in the graph. Thomassen has conjectured that a countable graph is bounded if and only if its edges can be oriented, possibly both ways, so that every vertex has finite out-degree and every ray has a forward oriented tail. We present a counterexample to this conjecture.

Analogues of the Erdős-Ko-Rado theorem are proved for the Boolean algebra of all subsets of {1,…n} and in this algebra truncated by the removal of the empty set and the whole set.

Gallai [1] raised the question of determining t(n), the maximum number of triangles in graphs of n vertices with acyclic neighborhoods. Here we disprove his conjecture (t(n) ~ n2/8) by exhibiting graphs having n2/7.5 triangles. We improve the upper bound [11] of (n2 − n)/6 to t(n) ≤; n2/7.02 + O(n). For regular graphs, we further decrease this bound to n2/7.75 + O(n).

A Steiner quadruple system SQS(v) of order v is a family ℬ of 4-element subsets of a v-element set V such that each 3-element subset of V is contained in precisely one B ∈ ℬ. We prove that if T ∩ B ≠ ø for all B ∈ ℬ (i.e., if T is a transversal), then |T| ≥ v/2, and if T is a transversal of cardinality exactly v/2, then V \ T is a transversal as well (i.e., T is a blocking set). Also, in respect of the so-called ‘doubling construction’ that produces SQS(2v) from two copies of SQS(v), we give a necessary and sufficient condition for this operation to yield a Steiner quadruple system with blocking sets.

Every general graph with degrees 2k and 2k − 2, k ≥ 3, with zero or at least two vertices of degree 2k − 2 in each component, has a k-edge-colouring such that each monochromatic subgraph has degree 1 or 2 at every vertex.

In particular, if T is a triangle in a 6-regular general graph, there exists a 2-factorization of G such that each factor uses an edge in T if and only if T is non-separating.

The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph Gnm are shown to be asymptotically normal if m is neither too large nor too small. At the lowest limit m ≍ n3/2, these numbers are asymptotically log-normal. For Gnp, the numbers are asymptotically log-normal for a wide range of p, including p constant. The same results are obtained for random directed graphs and bipartite graphs. The results are proved using decomposition and projection methods.

We show that, for any finite set P of points in the plane and for any integer k ≥ 2, there is a finite set R = R(P, k) with the following property: for any k-colouring of R there is a monochromatic set , ⊆ R, such that is combinatorially equivalent to the set P, and the convex hull of P contains no point of R \ . We also consider related questions for colourings of p-element subsets of R (p > 1), and show that these analogues have negative solutions.

It is shown that unless NP collapses to random polynomial time RP, there can be no fully polynomial randomised approximation scheme for the antiferromagnetic version of the Q-state Potts model.