2,571 research outputs found
Complex spherical codes with two inner products
A finite set in a complex sphere is called a complex spherical -code
if the number of inner products between two distinct vectors in is equal to
. In this paper, we characterize the tight complex spherical -codes by
doubly regular tournaments, or skew Hadamard matrices. We also give certain
maximal 2-codes relating to skew-symmetric -optimal designs. To prove them,
we show the smallest embedding dimension of a tournament into a complex sphere
by the multiplicity of the smallest or second-smallest eigenvalue of the Seidel
matrix.Comment: 10 pages, to appear in European Journal of Combinatoric
Embedding Four-directional Paths on Convex Point Sets
A directed path whose edges are assigned labels "up", "down", "right", or
"left" is called \emph{four-directional}, and \emph{three-directional} if at
most three out of the four labels are used. A \emph{direction-consistent
embedding} of an \mbox{-vertex} four-directional path on a set of
points in the plane is a straight-line drawing of where each vertex of
is mapped to a distinct point of and every edge points to the direction
specified by its label. We study planar direction-consistent embeddings of
three- and four-directional paths and provide a complete picture of the problem
for convex point sets.Comment: 11 pages, full conference version including all proof
Directed Ramsey number for trees
In this paper, we study Ramsey-type problems for directed graphs. We first
consider the -colour oriented Ramsey number of , denoted by
, which is the least for which every
-edge-coloured tournament on vertices contains a monochromatic copy of
. We prove that for any oriented
tree . This is a generalisation of a similar result for directed paths by
Chv\'atal and by Gy\'arf\'as and Lehel, and answers a question of Yuster. In
general, it is tight up to a constant factor.
We also consider the -colour directed Ramsey number
of , which is defined as above, but, instead
of colouring tournaments, we colour the complete directed graph of order .
Here we show that for any
oriented tree , which is again tight up to a constant factor, and it
generalises a result by Williamson and by Gy\'arf\'as and Lehel who determined
the -colour directed Ramsey number of directed paths.Comment: 27 pages, 14 figure
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