15,628 research outputs found
Low-distortion embeddings of graphs with large girth
The main purpose of the paper is to construct a sequence of graphs of
constant degree with indefinitely growing girths admitting embeddings into
with uniformly bounded distortions. This result answers the problem
posed by N. Linial, A. Magen, and A. Naor (2002).Comment: Some confusing omissions are corrected in the second versio
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
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