22,375 research outputs found
Long path and cycle decompositions of even hypercubes
We consider edge decompositions of the -dimensional hypercube into
isomorphic copies of a given graph . While a number of results are known
about decomposing into graphs from various classes, the simplest cases of
paths and cycles of a given length are far from being understood. A conjecture
of Erde asserts that if is even, and divides the number
of edges of , then the path of length decomposes . Tapadia et
al.\ proved that any path of length , where , satisfying these
conditions decomposes . Here, we make progress toward resolving Erde's
conjecture by showing that cycles of certain lengths up to
decompose . As a consequence, we show that can be decomposed into
copies of any path of length at most dividing the number of edges of
, thereby settling Erde's conjecture up to a linear factor
Phylogenetic toric varieties on graphs
We define phylogenetic projective toric model of a trivalent graph as a
generalization of a binary symmetric model of a trivalent phylogenetic tree.
Generators of the pro- jective coordinate ring of the models of graphs with one
cycle are explicitly described. The phylogenetic models of graphs with the same
topological invariants are deforma- tion equivalent and share the same Hilbert
function. We also provide an algorithm to compute the Hilbert function.Comment: 36 pages, improved expositio
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