435 research outputs found
Upper bound theorem for odd-dimensional flag triangulations of manifolds
We prove that among all flag triangulations of manifolds of odd dimension
2r-1 with sufficiently many vertices the unique maximizer of the entries of the
f-, h-, g- and gamma-vector is the balanced join of r cycles. Our proof uses
methods from extremal graph theory.Comment: Clarifications and new references, title has change
Rank of divisors on tropical curves
We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, we confirm a conjecture of Baker asserting that the rank of a divisor D on a (non-metric) graph is equal to the rank of D on the corresponding metric graph, and construct an algorithm for computing the rank of a divisor on a tropical curve
Loebl-Komlos-Sos Conjecture: dense case
We prove a version of the Loebl-Komlos-Sos Conjecture for dense graphs. For
each q>0 there exists a number such that for any n>n_0 and
k>qn the following holds: if G be a graph of order n with at least n/2 vertices
of degree at least k, then any tree of order k+1 is a subgraph of G.Comment: 56 pages, 8 figures; substantial changes as suggested by a refere
A density Corr\'adi-Hajnal Theorem
We find, for all sufficiently large and each , the maximum number of
edges in an -vertex graph which does not contain vertex-disjoint
triangles.
This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in
turn an extension of Mantel's Theorem. Our result can also be viewed as a
density version of the Corradi-Hajnal Theorem.Comment: 41 pages (including 11 pages of appendix), 4 figures, 2 table
An extension of Tur\'an's Theorem, uniqueness and stability
We determine the maximum number of edges of an -vertex graph with the
property that none of its -cliques intersects a fixed set .
For , the -partite Turan graph turns out to be the unique
extremal graph. For , there is a whole family of extremal graphs,
which we describe explicitly. In addition we provide corresponding stability
results.Comment: 12 pages, 1 figure; outline of the proof added and other referee's
comments incorporate
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