9 research outputs found
Spanning embeddings of arrangeable graphs with sublinear bandwidth
The Bandwidth Theorem of B\"ottcher, Schacht and Taraz [Mathematische Annalen
343 (1), 175-205] gives minimum degree conditions for the containment of
spanning graphs H with small bandwidth and bounded maximum degree. We
generalise this result to a-arrangeable graphs H with \Delta(H)<sqrt(n)/log(n),
where n is the number of vertices of H.
Our result implies that sufficiently large n-vertex graphs G with minimum
degree at least (3/4+\gamma)n contain almost all planar graphs on n vertices as
subgraphs. Using techniques developed by Allen, Brightwell and Skokan
[Combinatorica, to appear] we can also apply our methods to show that almost
all planar graphs H have Ramsey number at most 12|H|. We obtain corresponding
results for graphs embeddable on different orientable surfaces.Comment: 20 page
Local resilience of spanning subgraphs in sparse random graphs
For each real γ>0γ>0 and integers Δ≥2Δ≥2 and k≥1k≥1, we prove that there exist constants β>0β>0 and C>0C>0 such that for all p≥C(logn/n)1/Δp≥C(logn/n)1/Δ the random graph G(n,p)G(n,p) asymptotically almost surely contains – even after an adversary deletes an arbitrary (1/k−γ1/k−γ)-fraction of the edges at every vertex – a copy of every n-vertex graph with maximum degree at most Δ, bandwidth at most βn and at least Cmax{p−2,p−1logn}Cmax{p−2,p−1logn} vertices not in triangles
The bandwidth theorem for locally dense graphs
The Bandwidth theorem of B\"ottcher, Schacht and Taraz gives a condition on
the minimum degree of an -vertex graph that ensures contains every
-chromatic graph on vertices of bounded degree and of bandwidth
, thereby proving a conjecture of Bollob\'as and Koml\'os. In this paper
we prove a version of the Bandwidth theorem for locally dense graphs. Indeed,
we prove that every locally dense -vertex graph with contains as a subgraph any given (spanning) with bounded
maximum degree and sublinear bandwidth.Comment: 35 pages. Author accepted version, to appear in Forum of Mathematics,
Sigm
The bandwidth theorem in sparse graphs
The bandwidth theorem [Mathematische Annalen, 343(1):175–205, 2009] states that any n-vertex graph G with minimum degree [Formula Presented] contains all n-vertex k-colourable graphs H with bounded maximum degree and bandwidth o(n). We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs. More precisely, we show that for p ≫[Formula Presented] asymptotically almost surely each spanning subgraph G of G(n, p) with minimum degree [Formula Presented] pn contains all n-vertex k-colourable graphs H with maximum degree ∆, bandwidth o(n), and at least Cp−2 vertices not contained in any triangle. A similar result is shown for sufficiently bijumbled graphs, which, to the best of our knowledge, is the first resilience result in pseudorandom graphs for a rich class of spanning subgraphs. Finally, we provide improved results for H with small degeneracy, which in particular imply a resilience result in G(n, p) with respect to the containment of spanning bounded degree trees for p ≫[Formula Presented]
Large structures in dense directed graphs
We answer questions in extremal combinatorics, for directed graphs. Specifically, we investigate which large tree-like directed graphs are contained in all dense directed graphs of large order. More precisely, let T be an oriented tree of order n; among others, we establish the following results.
(1) We obtain a sufficient condition which ensures every tournament of order n contains T, and show that almost every tree possesses this property.
(2) We prove that for all positive C, ɛ and sufficiently large n, every tournament of order (1+ɛ)n contains T if Δ(T)≤(log n)^C.
(3) We prove that for all positive Δ, ɛ and sufficiently large n, every directed graph G of order n and minimum semidegree (1/2+ɛ)n contains T if Δ(T)≤Δ.
(4) We obtain a sufficient condition which ensures that every directed graph G of order n with minimum semidegree at least (1/2+ɛ)n contains T, and show that almost every tree possesses this property.
(5) We extend our method in (4) to a class of tree-like spanning graphs which includes all orientations of Hamilton cycles and large subdivisions of any graph.
Result (1) confirms a conjecture of Bender and Wormald and settles a conjecture of Havet and Thomassé for almost every tree; (2) strengthens a result of Kühn, Mycroft and Osthus; (3) is a directed graph analogue of a classical result of Komlós, Sárközy and Szemerédi and is implied by (4) and (5) is of independent interest