12,166 research outputs found
Packing and covering with balls on Busemann surfaces
In this note we prove that for any compact subset of a Busemann surface
(in particular, for any simple polygon with geodesic metric)
and any positive number , the minimum number of closed balls of radius
with centers at and covering the set is at most 19
times the maximum number of disjoint closed balls of radius centered
at points of : , where and
are the covering and the packing numbers of by -balls.Comment: 27 page
Monochromatic Clique Decompositions of Graphs
Let be a graph whose edges are coloured with colours, and be a -tuple of graphs. A monochromatic -decomposition of is a partition of the edge set of such that each
part is either a single edge or forms a monochromatic copy of in colour
, for some . Let be the smallest
number , such that, for every order- graph and every
-edge-colouring, there is a monochromatic -decomposition with at
most elements. Extending the previous results of Liu and Sousa
["Monochromatic -decompositions of graphs", Journal of Graph Theory},
76:89--100, 2014], we solve this problem when each graph in is a
clique and is sufficiently large.Comment: 14 pages; to appear in J Graph Theor
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