A digraph D is said to be an m-coloured digraph, if its arcs 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) of vertices of D is said to be a kernel by monochromatic paths of the m-coloured digraph D, if it satisfies the two following properties : (1) N is independent by monochromatic paths; i.e. f...

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 directed cycle is called quasi-monochromatic if with at most one exception 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 following t...

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

LEMMA . At most (kn)'g"ulk k-sets Sc [ n ] have IP(S)I <u . Proof. Let a,< . . . < a k denote the elements of S. Call i doubling if P(a, , . . ., a i) has double the size of P(a, , . . ., a i 1 ) . There are at most lg u doubling i. Hence there are at most k'g" choices for doubling positions i and at most n'g" choices for the values a i . If i is not doubling then a i = x y, where x, y E P(a, ,...

We show that for any colouring of the edges of the complete bipartite graphKn,n with 3 colours there are 5 disjoint monochromatic cycles which together cover all but o(n) of the vertices and 18 disjoint monochromatic cycles which together cover all vertices. keywords: monochromatic cycle partition, Ramsey-type problem, complete bipartite graph MSC 2010: 05C69, 05C75, 05C38.

Here we address the problem to partition edge colored hypergraphs by monochromatic paths and cycles generalizing a well-known similar problem for graphs. We show that r-colored r-uniform complete hypergraphs can be partitioned into monochromatic Berge-paths of distinct colors. Also, apart from 2k − 5 vertices, 2-colored k-uniform hypergraphs can be partitioned into two monochromatic loose paths...

We call the digraph D an m-colored digraph if the arcs of D are colored with m colors. A path (or a cycle) is called monochromatic if all of its arcs are colored alike. A cycle is called a quasi-monochromatic cycle if with at most one exception all of its arcs are colored alike. A subdigraph H in D is called rainbow if all its arcs have different colors. A set N ⊆ V (D) is said to be a kernel b...