2 research outputs found
On the number of hypercubic bipartitions of an integer
We revisit a well-known divide-and-conquer maximin recurrence where the maximum is taken over all
proper bipartitions , and we present a new characterization of the
pairs summing to that yield the maximum . This new characterization allows us, for a given n\in\nats,
to determine the number of these bipartitions that yield the said
maximum . We present recursive formulae for , a generating function
, and an explicit formula for in terms of a special representation
of .Comment: 13 page
Induced subgraphs of hypercubes
Let denote the -dimensional hypercube on vertices. A vertex in
a subgraph of is {\em full} if its degree is . We apply the
Kruskal-Katona Theorem to compute the maximum number of full vertices an
induced subgraph on vertices of can have, as a function of
and . This is then used to determine
where (i) and are induced subgraphs of , and (ii) together
they cover all the edges of , that is .Comment: 16 page