We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours. A directed path (or a directed cycle) is called monochromatic if all of its arcs are coloured alike. A set N ⊆ V (D) is said to be a kernel by monochromatic paths if it satisfies the two following conditions (i) for every pair of different vertices u, v ∈ N there is no monochromatic directed path between t...

Fix c : [N] → r, an r-coloring of the pairs of natural numbers. An ordered list of distinct integers, a0, a1, a2 . . . ai−1, ai, ai+1 . . . is a monochromatic path for color k, if, for all i ≥ 1, c({ai−1, ai}) = k. The empty list is considered a path of any color k. Similarly the list of one element, a0, is also considered a path of any color k. For any monochromatic path of length two or more ...

We call the digraph D an m-coloured digraph if the arcs of D are coloured with m colours and all of them are used. A directed path is called monochromatic if all of its arcs are coloured alike. A set N of vertices of D is called a kernel by monochromatic paths if for every pair of vertices there is no monochromatic path between them and for every vertex v in V (D) \ N there is a monochromatic p...

BACKGROUND An optimal metal artifact reduction (MAR) technique is needed for a reliable and accurate image-based diagnosis. PURPOSE Using a 320-row scanner, we compared the dual-energy computed tomography (CT)-based monochromatic and the single-energy metal artifact reduction (SEMAR) techniques for CT venography (CTV) to identify the better imaging method for diagnosing deep vein thrombosis (...

In this note we report on our recent work, still in progress, regarding Folkman numbers. Let f(2, 3, 4) denote the smallest integer n such that there exists a K4– free graph of order n having that property that any 2–coloring of its edges yields at least one monochromatic triangle. It is well–known that such a number must exist [4,10]. For almost twenty years the best known upper bound, given b...

For fixed positive integers r, k and ` with 1 ≤ ` < r and an r-uniform hypergraph H, let κ(H, k, `) denote the number of k-colorings of the set of hyperedges of H for which any two hyperedges in the same color class intersect in at least ` elements. Consider the function KC(n, r, k, `) = maxH∈Hn κ(H, k, `), where the maximum runs over the family Hn of all r-uniform hypergraphs on n vertices. In...

A Folkman graph is a K4-free graph G such that if the edges of G are 2-colored, then there exists a monochromatic triangle. Erdős offered a prize for proving the existence of a Folkman graph with at most 1 million vertices. In this paper, we construct several “small” Folkman graphs within this limit. In particular, there exists a Folkman graph on 9697 vertices.

For xed positive integers r, k and ` with ` < r, and an r-uniform hypergraph H, let κ(H, k, `) denote the number of k-colorings of the set of hyperedges of H for which any two hyperedges in the same color class intersect in at least ` vertices. Consider the function KC(n, r, k, `) = maxH∈Hn κ(H, k, `), where the maximum runs over the family Hn of all r-uniform hypergraphs on n vertices. In this...