1,120 research outputs found
Orientations making k-cycles cyclic
We show that the minimum number of orientations of the edges of the n-vertex
complete graph having the property that every triangle is made cyclic in at
least one of them is . More generally, we also
determine the minimum number of orientations of such that at least one of
them orients some specific -cycles cyclically on every -element subset of
the vertex set. The questions answered by these results were motivated by an
analogous problem of Vera T. S\'os concerning triangles and -edge-colorings.
Some variants of the problem are also considered.Comment: 9 page
Families of graph-different Hamilton paths
Let D be an arbitrary subset of the natural numbers. For every n, let M(n;D)
be the maximum of the cardinality of a set of Hamiltonian paths in the complete
graph K_n such that the union of any two paths from the family contains a not
necessarily induced cycle of some length from D. We determine or bound the
asymptotics of M(n;D) in various special cases. This problem is closely related
to that of the permutation capacity of graphs and constitutes a further
extension of the problem area around Shannon capacity. We also discuss how to
generalize our cycle-difference problems and present an example where cycles
are replaced by 4-cliques. These problems are in a natural duality to those of
graph intersection, initiated by Erd\"os, Simonovits and S\'os. The lack of
kernel structure as a natural candidate for optimum makes our problems quite
challenging
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