46,810 research outputs found
The exact minimum number of triangles in graphs of given order and size
What is the minimum number of triangles in a graph of given order and size?
Motivated by earlier results of Mantel and Tur\'an, Rademacher solved the first
non-trivial case of this problem in 1941. The problem was revived by Erd\H{o}s
in 1955; it is now known as the Erd\H{o}s-Rademacher problem. After attracting
much attention, it was solved asymptotically in a major breakthrough by
Razborov in 2008. In this paper, we provide an exact solution for all large
graphs whose edge density is bounded away from~, which in this range
confirms a conjecture of Lov\'asz and Simonovits from 1975. Furthermore, we
give a description of the extremal graphs.Comment: Published in Forum of Mathematics, Pi, Volume 8, e8 (2020
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
Many copies in -free graphs
For two graphs and with no isolated vertices and for an integer ,
let denote the maximum possible number of copies of in an
-free graph on vertices. The study of this function when is a
single edge is the main subject of extremal graph theory. In the present paper
we investigate the general function, focusing on the cases of triangles,
complete graphs, complete bipartite graphs and trees. These cases reveal
several interesting phenomena. Three representative results are:
(i)
(ii) For any fixed , and ,
and
(iii) For any two trees and , where
is an integer depending on and (its precise definition is
given in Section 1).
The first result improves (slightly) an estimate of Bollob\'as and Gy\H{o}ri.
The proofs combine combinatorial and probabilistic arguments with simple
spectral techniques
Embedding graphs having Ore-degree at most five
Let and be graphs on vertices, where is sufficiently large.
We prove that if has Ore-degree at most 5 and has minimum degree at
least then Comment: accepted for publication at SIAM J. Disc. Mat
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