65,838 research outputs found
Triangle-Intersecting Families of Graphs
A family of graphs F is said to be triangle-intersecting if for any two
graphs G,H in F, the intersection of G and H contains a triangle. A conjecture
of Simonovits and Sos from 1976 states that the largest triangle-intersecting
families of graphs on a fixed set of n vertices are those obtained by fixing a
specific triangle and taking all graphs containing it, resulting in a family of
size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations
(for example, we prove that the same is true of odd-cycle-intersecting
families, and we obtain best possible bounds on the size of the family under
different, not necessarily uniform, measures). We also obtain stability
results, showing that almost-largest triangle-intersecting families have
approximately the same structure.Comment: 43 page
A better proof of the Goldman-Parker conjecture
The Goldman-Parker Conjecture classifies the complex hyperbolic C-reflection
ideal triangle groups up to discreteness. We proved the Goldman-Parker
Conjecture in [Ann. of Math. 153 (2001) 533--598] using a rigorous
computer-assisted proof. In this paper we give a new and improved proof of the
Goldman-Parker Conjecture. While the proof relies on the computer for extensive
guidance, the proof itself is traditional.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol9/paper35.abs.htm
Torsion and ground state maxima: close but not the same
Could the location of the maximum point for a positive solution of a
semilinear Poisson equation on a convex domain be independent of the form of
the nonlinearity? Cima and Derrick found certain evidence for this surprising
conjecture.
We construct counterexamples on the half-disk, by working with the torsion
function and first Dirichlet eigenfunction. On an isosceles right triangle the
conjecture fails again. Yet the conjecture has merit, since the maxima of the
torsion function and eigenfunction are unexpectedly close together. It is an
open problem to quantify this closeness in terms of the domain and the
nonlinearity
Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles
Given a simple graph , a subset of is called a triangle cover if
it intersects each triangle of . Let and denote the
maximum number of pairwise edge-disjoint triangles in and the minimum
cardinality of a triangle cover of , respectively. Tuza conjectured in 1981
that holds for every graph . In this paper, using a
hypergraph approach, we design polynomial-time combinatorial algorithms for
finding small triangle covers. These algorithms imply new sufficient conditions
for Tuza's conjecture on covering and packing triangles. More precisely,
suppose that the set of triangles covers all edges in . We
show that a triangle cover of with cardinality at most can be
found in polynomial time if one of the following conditions is satisfied: (i)
, (ii) , (iii)
.
Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs,
Combinatorial algorithm
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