A variation of Menger's theorem for long paths

In this paper we prove a Mengerian theorem for long paths, namely, that if in order to cut every uv-path of length at least n (n ≥ 2), in a diagraph D, we need to remove at least h points, then there exist {$\text{h}\text{(3n ? 5)}$} interior disjoint uv-paths in D of length at least n. Some variations and applications of this theorem are given as well.     

An Anti-Ramsey Theorem on Cycles

Let h(n,p) be the minimum integer such that every edge-colouring of the complete graph of order n, using exactly h(n,p) colours, produces at least one cycle of order p having all its edges of different colours. In this paper the value of h(n,p) is determinated for n≥p≥3. As a corollary we obtain the equality which was conjectured by Erdös, Simonovits and Sós, 30 years ago [4].     

The dichromatic number of a digraph

In this paper the concept of dichromatic number of a digraph which is a generalization of the chromatic number of a graph is introduced. The dichromatic number of a digraph D is defined as the minimum number of colours required to colour the vertices of D in such a way that the chromatic classes induce acyclic subdigraphs in D. Some results relating the dichromatic number of D with the existence of cycles of special lengths in D are presented. Contributions to chromatic theory are also obtained. In particular, we generalize the theorem due to P. Erdös and A. Hajnal (Acta Math. Acad. Sci. Hungar.17 (1966), 61–99) which states the existence of odd cycles of length ≥_χ(G) ? 1 in any graph G.     

Compatible connectedness in graphs and topological spaces

A topology on the vertex set of a graphG iscompatible with the graph if every induced subgraph ofG is connected if and only if its vertex set is topologically connected. In the case of locally finite graphs with a finite number of components, it was shown in [11] that a compatible topology exists if and only if the graph is a comparability graph and that all such topologies are Alexandroff. The main results of Section 1 extend these results to a much wider class of graphs. In Section 2, we obtain sufficient conditions on a graph under which all the compatible topologies are Alexandroff and in the case of bipartite graphs we show that this condition is also necessary.     

An Anti-Ramsey Theorem

Let be the Turán number which gives the maximum size of a graph of order containing no subgraph isomorphic to . In 1973, Erdős, Simonovits and Sós [5] proved the existence of an integer such that for every integer , the minimum number of colours , such that every -colouring of the edges of which uses all the colours produces at least one all whose edges have different colours, is given by . However, no estimation of was given in [5]. In this paper we prove that for . This formula covers all the relevant values of n and p. Received January 27, 1997/Revised March 14, 2000     

Two Order Invariants Related to the Fixed Point Property

We introduce two numerical invariants of orders that measure how close a poset is to having the fixed point property. We give general properties of those invariants and link them to known results on the fixed point property.     

Fixed points of posets and clique graphs

The fixed point property for partial orders has been the object of much attention in the past twenty years. Recently, M. Roddy ([7]) proved this famous conjecture of Rival (see [6]): the class of finite orders with the fixed point property is closed under finite products.In this article, we prove that a finite order has the fixed point property if the sequence of iterated clique graphs of its comparability graph tends to the trivial graph.     

Compact Compatible Topologies for Posets and Graphs

A topology on the vertex set of a comparability graph G is said to be compatible (respectively, weakly compatible) with G if each induced subgraph (respectively, each finite induced subgraph) is topologically connected if and only it it is graph-connected; a weakly compatible topology on the vertex set of a graph completely determines the graph structure. We consider here the problem of deciding whether or not a comparability graph has a compact compatible or weakly compatible topology and in the case of graphs with small cycles, hence in the case of trees, we give a characterization.