Let rex(n,F) denote the maximum number of edges in an n-vertex graph that is regular and does not contain F as a subgraph. We give lower bounds on rex(n,F), are best possible up to constant factor, when one C4, K2,t, K3,3 or Ks,t for t>s!.

We characterize the graphs for which all 2-connected non-bipartite subgraphs have a strongly connected orientation in which each directed circuit has an odd number of edges. We also give a polynomial-time algorithm to find such an orientation in these graphs. Moreover, we give an algorithm that given any orientation of such a graph, determines if it has an even directed circuit. The proofs of t...

We investigate the following generalisation of the ‘multiplication table problem’ of Erdős: given a bipartite graph with m edges, how large is the set of sizes of its induced subgraphs? Erdős’s problem, posed in 1955, of estimating the number of distinct products ab with a, b ≤ n is precisely the problem under consideration when the graph in question is the complete bipartite graph Kn,n. In thi...

A graph is called supermagic if it admits a labelling of the edges by pairwise different consecutive positive integers such that the sum of the labels of the edges incident with a vertex is independent of the particular vertex. In this paper we will introduce a new construction of supermagic graphs using the special structure of their subgraphs. The supermagic complements of some bipartite grap...

A bipartite graph on 2n vertices is bipancyclic if it contains cycles of all even lengths from 4 to 2n. In this paper we prove that the random bipartite graph G(n, n, p) with p(n) À n−2/3 asymptotically almost surely has the following resilience property: Every Hamiltonian subgraph G′ of G(n, n, p) with more than (1/2 + o(1))n2p edges is bipancyclic. This result is tight in two ways. First, the...

By using the notion of compatibility of subgraphs with a perfect matching developed for digraphs in [1], we show that if, in a balanced bipartite graph G of minimum degree 6, the maximum cardinality ebip of a balanced independent subset satisfies ~bip ~< 26-4, then G is hamiltonian-biconnected, and if Ctbip ~< 26-2, G contains a hamiltonian path. Moreover, we give some properties of balanced bi...