Let k and ` be positive integers. With a graph G, we associate the quantity ck,`(G), the number of k-colorings of the edge set of G with no monochromatic matching of size `. Consider the function ck,` : N −→ N given by ck,`(n) = max {ck,`(G) : |V (G)| = n}, the maximum of ck,`(G) over all graphs G on n vertices. In this paper, we determine ck,`(n) and the corresponding extremal graphs for all l...

We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with outdegrees prescribed by a function α : V → N unifies many different combinatorial structures, including the afore mentioned. We call these orientations α-orien...

We study a variation of the assignment problem in operations research and formulate it in terms of graphs as follows. Suppose G = (V; E) is a graph and U a subset of V. A-assignment of G with respect to U is an edge set X such that deg X (v) = 1 for all vertices v in U , where deg X (v) is the degree of v in the subgraph of G induced by the edge set X. The-assignment problem is to nd a-assignme...

Matching is a basic operation extensively used in computation. Second-order matching, in particular, provides an adequate environment for expressing program transformations and pattern recognition for automated deduction. The past few years have established the benefit of using explicit substitutions for theorem proving and higher-order unification. In this paper, we will make use of explicit s...

ontology is the main infrastructure of the semantic web which provides facilities for integration, searching and sharing of information on the web. development of ontologies as the basis of semantic web and their heterogeneities have led to the existence of ontology matching. by emerging large-scale ontologies in real domain, the ontology matching systems faced with some problem like memory con...

Eroh and Oellerman defined BRR(G1, G2) as the smallest N such that any edge coloring of the complete bipartite graph KN,N contains either a monochromatic G1 or a multicolored G2. We restate the problem of determining BRR(K1,λ,Kr,s) in matrix form and prove estimates and exact values for several choices of the parameters. Our general bound uses Füredi’s result on fractional matchings of uniform ...

For a k-uniform hypergraph G with vertex set {1, . . . , n}, the ordered Ramsey number ORt(G) is the least integer N such that every t-coloring of the edges of the complete k-uniform graph on vertex set {1, . . . , N} contains a monochromatic copy of G whose vertices follow the prescribed order. Due to this added order restriction, the ordered Ramsey numbers can be much larger than the usual gr...

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by n colours with at least n + 1 edges of each colour there must be a matching that uses each colour exactly once. In this paper we consider the same question without the bipartiteness assumption. We show that in any multigraph with edge multiplicities o(n) that is properly edge-coloured by n colours ...

V := {ri}i=1 ∪ {ci}i=1 and E := {{ri, cj} | mi,j ≥ 1/n}. The main observation is that if G contains a perfect matching then there exists a desired permutation. To see that, let us assume that P = {e1, . . . , en} is a subset of the edges of G which form a perfect matching, with ei = {ri, cji}. Then setting π(i) := ji satisfies the desired properties: since P is a matching we have that π is a bi...

The cell rotation graph D(G) on the strongly connected orientations of a 2-edge-connected plane graph G is de5ned. It is shown that D(G) is a directed forest and every component is an in-tree with one root; if T is a component of D(G), the reversions of all orientations in T induce a component of D(G), denoted by T−, thus (T; T−) is called a pair of in-trees of D(G); G is Eulerian if and only i...