18,871 research outputs found
A graph partition problem
Given a graph on vertices, for which is it possible to partition
the edge set of the -fold complete graph into copies of ? We show
that there is an integer , which we call the \emph{partition modulus of
}, such that the set of values of for which such a partition
exists consists of all but finitely many multiples of . Trivial
divisibility conditions derived from give an integer which divides
; we call the quotient the \emph{partition index of }. It
seems that most graphs have partition index equal to , but we give two
infinite families of graphs for which this is not true. We also compute
for various graphs, and outline some connections between our problem and the
existence of designs of various types
2178 And All That
For integers g >= 3, k >= 2, call a number N a (g,k)-reverse multiple if the
reversal of N in base g is equal to k times N. The numbers 1089 and 2178 are
the two smallest (10,k)-reverse multiples, their reversals being 9801 = 9x1089
and 8712 = 4x2178. In 1992, A. L. Young introduced certain trees in order to
study the problem of finding all (g,k)-reverse multiples. By using modified
versions of her trees, which we call Young graphs, we determine the possible
values of k for bases g = 2 through 100, and then show how to apply the
transfer-matrix method to enumerate the (g,k)-reverse multiples with a given
number of base-g digits. These Young graphs are interesting finite directed
graphs, whose structure is not at all well understood.Comment: 22 pages, 16 figures, one table. July 4 2013: corrected typo in
table, added conjectures about particular graphs. Sept. 24 2013: corrected
typos, added conjectures and theorems. Oct 13 2013: minor edit
Classifying Voronoi graphs of hex spheres
A hex sphere is a singular Euclidean sphere with four cones points whose cone
angles are (integer) multiples of 2*pi/3 but less than 2*pi. Given a hex sphere
M, we consider its Voronoi decomposition centered at the two cone points with
greatest cone angles. In this paper we use elementary Euclidean geometry to
describe the Voronoi regions of hex spheres and classify the Voronoi graphs of
hex spheres (up to graph isomorphism).Comment: 14 pages, 9 figure
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