15 research outputs found
Dense H-free graphs are almost (Χ(H)-1)-partite
By using the Szemeredi Regularity Lemma, Alon and Sudakov recently
extended the classical Andrasfai-Erdos-Sos theorem to cover general graphs. We
prove, without using the Regularity Lemma, that the following stronger statement
is true.
Given any (r+1)-partite graph H whose smallest part has t vertices, there exists
a constant C such that for any given Îľ>0 and sufficiently large n the following is
true. Whenever G is an n-vertex graph with minimum degree
δ(G)âĽ(1 â
3/3râ1 + Îľ)n,
either G contains H, or we can delete f(n,H)â¤Cn2â1/t edges from G to obtain an
r-partite graph. Further, we are able to determine the correct order of magnitude
of f(n,H) in terms of the Zarankiewicz extremal function
Triangle-free subgraphs of random graphs
Recently there has been much interest in studying random graph analogues of
well known classical results in extremal graph theory. Here we follow this
trend and investigate the structure of triangle-free subgraphs of with
high minimum degree. We prove that asymptotically almost surely each
triangle-free spanning subgraph of with minimum degree at least
is -close to bipartite,
and each spanning triangle-free subgraph of with minimum degree at
least is -close to
-partite for some . These are random graph analogues of a
result by Andr\'asfai, Erd\H{o}s, and S\'os [Discrete Math. 8 (1974), 205-218],
and a result by Thomassen [Combinatorica 22 (2002), 591--596]. We also show
that our results are best possible up to a constant factor.Comment: 18 page
Chromatic thresholds in dense random graphs
The chromatic threshold of a graph with respect to the
random graph is the infimum over such that the following holds
with high probability: the family of -free graphs with
minimum degree has bounded chromatic number. The study of
the parameter was initiated in 1973 by
Erd\H{o}s and Simonovits, and was recently determined for all graphs . In
this paper we show that for all fixed , but that typically if . We also make significant progress towards determining
for all graphs in the range . In sparser random graphs the
problem is somewhat more complicated, and is studied in a separate paper.Comment: 36 pages (including appendix), 1 figure; the appendix is copied with
minor modifications from arXiv:1108.1746 for a self-contained proof of a
technical lemma; accepted to Random Structures and Algorithm
Minimum degree stability of H-free graphs
Given an (r + 1)-chromatic graph H, the fundamental edge stability result of ErdĹs and Simonovits says that all n-vertex H-free graphs have at most (1 â 1/r + o(1))( n 2 ) edges, and any H-free graph with that many edges can be made r-partite by deleting o(n 2 ) edges. Here we consider a natural variant of this â the minimum degree stability of H-free graphs. In particular, what is the least c such that any n-vertex H-free graph with minimum degree greater than cn can be made r-partite by deleting o(n 2 ) edges? We determine this least value for all 3- chromatic H and for very many non-3-colourable H (all those in which one is commonly interested) as well as bounding it for the remainder. This extends the AndrĂĄsfai-ErdĹs-SĂłs theorem and work of Alon and Sudako
Triangle-free subgraphs of random graphs
Recently there has been much interest in studying random graph analogues of well known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of G(n, p) with high minimum degree. We prove that asymptotically almost surely each triangle-free spanning subgraph of G(n, p) with minimum degree at least ( 2 + o(1)lpn is O(pâ1 n)-close to bipartite, and each spanning triangle-free subgraph of G(n, p) with minimum degree at least ( 1 + Îľ)pn is O(pâ1 n)-close to r-partite for some r = r(Îľ). These are random graph analogues of a result by AndrĂĄsfai, ErdĹs and SĂłs [Discrete Math. 8 (1974), 205â218], and a result by Thomassen [Combinatorica 22 (2002), 591â596]. We also show that our results are best possible up to a constant factor