1,195 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
Large rainbow matchings in large graphs
A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose
edges have distinct colors. The \textit{color degree} of a vertex is the
number of different colors on edges incident to . We show that if is
large enough (namely, ), then each -vertex graph with
minimum color degree at least contains a rainbow matching of size at least
- …