129,981 research outputs found
Extension complexity of stable set polytopes of bipartite graphs
The extension complexity of a polytope is the minimum
number of facets of a polytope that affinely projects to . Let be a
bipartite graph with vertices, edges, and no isolated vertices. Let
be the convex hull of the stable sets of . It is easy to
see that . We improve
both of these bounds. For the upper bound, we show that is , which is an improvement when
has quadratically many edges. For the lower bound, we prove that
is when is the
incidence graph of a finite projective plane. We also provide examples of
-regular bipartite graphs such that the edge vs stable set matrix of
has a fooling set of size .Comment: 13 pages, 2 figure
The Maximum Wiener Index of Maximal Planar Graphs
The Wiener index of a connected graph is the sum of the distances between all
pairs of vertices in the graph. It was conjectured that the Wiener index of an
-vertex maximal planar graph is at most
. We prove this conjecture and for every
, , determine the unique -vertex maximal planar graph for
which this maximum is attained.Comment: 13 pages, 4 figure
Maximal partial spreads and the modular n-queen problem III
AbstractMaximal partial spreads in PG(3,q)q=pk,p odd prime and q⩾7, are constructed for any integer n in the interval (q2+1)/2+6⩽n⩽(5q2+4q−1)/8 in the case q+1≡0,±2,±4,±6,±10,12(mod24). In all these cases, maximal partial spreads of the size (q2+1)/2+n have also been constructed for some small values of the integer n. These values depend on q and are mainly n=3 and n=4. Combining these results with previous results of the author and with that of others we can conclude that there exist maximal partial spreads in PG(3,q),q=pk where p is an odd prime and q⩾7, of size n for any integer n in the interval (q2+1)/2+6⩽n⩽q2−q+2
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