4 research outputs found
Local colourings and monochromatic partitions in complete bipartite graphs
We show that for any -local colouring of the edges of the balanced
complete bipartite graph , its vertices can be covered with at
most~ disjoint monochromatic paths. And, we can cover almost all vertices of
any complete or balanced complete bipartite -locally coloured graph with
disjoint monochromatic cycles.\\ We also determine the -local
bipartite Ramsey number of a path almost exactly: Every -local colouring of
the edges of contains a monochromatic path on vertices.Comment: 18 page
The threshold for the constrained Ramsey property
Given graphs , , and , let
denote the property that in every edge colouring of there is a
monochromatic copy of or a rainbow copy of . The constrained Ramsey
number, defined as the minimum such that
, exists if and only if is a star or
is a forest. We determine the threshold for the property
when is a forest, explicitly
when the threshold is and implicitly otherwise.Comment: 15 page