For every graph G, let σ2(G) = min{d(x)+ d(y) : xy 6∈ E(G)}. The main result of the paper says that everyn-vertex graphGwithσ2(G) ≥ 4n 3 −1 contains each spanning subgraphH all whose components are isomorphic to graphs in {K1, K2, C3, K 4 , C + 5 }. This generalizes the earlier results of Justesen, Enomoto, andWang, and is a step towards an Ore-type analogue of the Bollobás–Eldridge–Catlin Conj...

Consider two graphs G1 and G2 on the same vertex set V and suppose that Gi has mi edges. Then there is a bipartition of V into two classes A and B so that for both i = 1, 2 we have eGi(A,B) ≥ mi/2 − √ mi. This answers a question of Bollobás and Scott. We also prove results about partitions into more than two vertex classes.

A graph G is said to be Cl-saturated if G contains no cycle of length l, but for any edge in the complement of G the graph G + e does contain a cycle of length l. The minimum number of edges of a Cl-saturated graph was shown by Barefoot et al. to be between n + c1 n l and n + c2 n l for some positive constants c1 and c2. This confirmed a conjecture of Bollobás. Here we improve the value of c2 f...

Let G be an abelian group of order k. How is the problem of minimizing the number of sums from a sequence of given length in G related to the problem of minimizing the number of k-sums? In this paper we show that the minimum number of k-sums for a sequence a1, . . . , ar that does not have 0 as a k-sum is attained at the sequence b1, . . . , br−k+1,0, . . . ,0, where b1, . . . , br−k+1 is chose...

as was introduced by Bollobás, Janson and Riordan in [1]. Here S is a separable metric space and μ is a Borel probability measure on S . No relationship is assumed between x (n) i and x (n′) i . To simplify notation we shall from now on write (x1, . . . , xn) = (x (n) 1 , . . . , x (n) n ). We begin by recalling some basic definitions and assumptions from [1]. For each n let (x1, . . . , xn) be...

We consider two operations on the edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of S3 e(G), the ribbon group of G, on G. We show that this ribbon group action gives a complete characterization of duality in that if G is any cellularly embedded graph wit...

We determine an Ore type condition that allows the embedding of 3-colourable bounded degree graphs of sublinear bandwidth: For all ∆, γ > 0 there are β, n0 > 0 such that for all n ≥ n0 the following holds. Let G = (V,E) and H be n-vertex graphs such that H is 3-colourable, has maximum degree ∆(H) ≤ ∆ and bandwidth bw(H) ≤ βn, and G satisfies deg(u) + deg(v) ≥ ( 3 + γ)n for all uv / ∈ E. Then G ...

Some parts of the material in this paper were presented by the author in the meeting Combinatorics 2008 celebrated at Costermano, Italy the last June. So it contains some expository on previous work but it also contains new and original results, mainly those in the last two sections. The results in this paper are consequences of a special interpretation as fixed point problems of the two classi...

For classes O of structures on finite linear orders (permutations, ordered graphs etc.) endowed with containment order (containment of permutations, subgraph relation etc.), we investigate restrictions on the function f(n) counting objects with size n in a lower ideal in (O, ). We present a framework of edge P -colored complete graphs (C(P ), ) which includes many of these situations, and we pr...

A cancellative hypergraph has no three edges A;B;C with ADBCC: We give a new short proof of an old result of Bollobás, which states that the maximum size of a cancellative triple system is achieved by the balanced complete tripartite 3-graph. One of the two forbidden subhypergraphs in a cancellative 3-graph is F5 1⁄4 fabc; abd; cdeg: For nX33 we show that the maximum number of triples on n vert...