75 research outputs found
Improper colourings inspired by Hadwigerâs conjecture
Hadwigerâs Conjecture asserts that every Kt-minor-free graph has a proper (t â 1)-colouring. We relax the conclusion in Hadwigerâs Conjecture via improper colourings. We prove that every Kt-minor-free graph is (2t â 2)-colourable with monochromatic components of order at most 1/2 (t â 2). This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every Kt-minor-free graph is (t â 1)-colourable with monochromatic degree at most t â 2. This is the best known degree bound for such a result. Both these theorems are based on a decomposition method of independent interest. We give analogous results for Ks,t-minorfree graphs, which lead to improved bounds on generalised colouring numbers for these classes. Finally, we prove that graphs containing no Kt-immersion are 2-colourable with bounded monochromatic degree
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
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
On sufficient conditions for Hamiltonicity in dense graphs
We study structural conditions in dense graphs that guarantee the existence
of vertex-spanning substructures such as Hamilton cycles. It is easy to see
that every Hamiltonian graph is connected, has a perfect fractional matching
and, excluding the bipartite case, contains an odd cycle. Our main result in
turn states that any large enough graph that robustly satisfies these
properties must already be Hamiltonian. Moreover, the same holds for embedding
powers of cycles and graphs of sublinear bandwidth subject to natural
generalisations of connectivity, matchings and odd cycles.
This solves the embedding problem that underlies multiple lines of research
on sufficient conditions for Hamiltonicity in dense graphs. As applications, we
recover and establish Bandwidth Theorems in a variety of settings including
Ore-type degree conditions, P\'osa-type degree conditions, deficiency-type
conditions, locally dense and inseparable graphs, multipartite graphs as well
as robust expanders
Local resilience of spanning subgraphs in sparse random graphs
For each real Îł>0Îł>0 and integers Îâ„2Îâ„2 and kâ„1kâ„1, we prove that there exist constants ÎČ>0ÎČ>0 and C>0C>0 such that for all pâ„C(logâĄn/n)1/Îpâ„C(logâĄn/n)1/Î the random graph G(n,p)G(n,p) asymptotically almost surely contains â even after an adversary deletes an arbitrary (1/kâÎł1/kâÎł)-fraction of the edges at every vertex â a copy of every n-vertex graph with maximum degree at most Î, bandwidth at most ÎČn and at least CmaxâĄ{pâ2,pâ1logâĄn}CmaxâĄ{pâ2,pâ1logâĄn} vertices not in triangles
Forcing spanning subgraphs via Ore type conditions
Abstract We determine an Ore type condition that allows the embedding of 3-colourable bounded degree graphs of sublinear bandwidth: For all â, Îł > 0 there are ÎČ, n 0 > 0 such that for all n â„ n 0 the following holds. Let G = (V, E) and H be n-vertex graphs such that H is 3-colourable, has maximum degree â(H) †â and bandwidth bw(H) †ÎČn, and G satisfies deg(u) + deg(v) â„
k-L(2,1)-Labelling for Planar Graphs is NP-Complete for k >= 4
A mapping from the vertex set of a graph G = (V,E) into an interval of
integers {0,...,k} is an L(2,1)-labelling of G of span k if any two adjacent
vertices are mapped onto integers that are at least 2 apart, and every two
vertices with a common neighbour are mapped onto distinct integers. It is known
that for any fixed k >= 4, deciding the existence of such a labelling is an
NP-complete problem while it is polynomial for k = 8, it
remains NP-complete when restricted to planar graphs. In this paper, we show
that it remains NP-complete for any k >= 4 by reduction from Planar Cubic
Two-Colourable Perfect Matching. Schaefer stated without proof that Planar
Cubic Two-Colourable Perfect Matching is NP-complete. In this paper we give a
proof of this.Comment: 16 pages, includes figures generated using PSTricks. To appear in
Discrete Applied Mathematics. Some very minor corrections incorporate
Embedding spanning bipartite graphs of small bandwidth
Boettcher, Schacht and Taraz gave a condition on the minimum degree of a
graph G on n vertices that ensures G contains every r-chromatic graph H on n
vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture
of Bollobas and Komlos. We strengthen this result in the case when H is
bipartite. Indeed, we give an essentially best-possible condition on the degree
sequence of a graph G on n vertices that forces G to contain every bipartite
graph H on n vertices of bounded degree and of bandwidth o(n). This also
implies an Ore-type result. In fact, we prove a much stronger result where the
condition on G is relaxed to a certain robust expansion property. Our result
also confirms the bipartite case of a conjecture of Balogh, Kostochka and
Treglown concerning the degree sequence of a graph which forces a perfect
H-packing.Comment: 23 pages, file updated, to appear in Combinatorics, Probability and
Computin
Embedding into bipartite graphs
The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher,
Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any
, every balanced bipartite graph on vertices with bounded degree
and sublinear bandwidth appears as a subgraph of any -vertex graph with
minimum degree , provided that is sufficiently large. We show
that this threshold can be cut in half to an essentially best-possible minimum
degree of when we have the additional structural
information of the host graph being balanced bipartite. This complements
results of Zhao [to appear in SIAM J. Discrete Math.], as well as Hladk\'y and
Schacht [to appear in SIAM J. Discrete Math.], who determined a corresponding
minimum degree threshold for -factors, with and fixed.
Moreover, it implies that the set of Hamilton cycles of is a generating
system for its cycle space.Comment: 16 pages, 2 figure
Ramsey numbers of squares of paths
The Ramsey number R(G;H) has been actively studied for the past 40 years, and it was determined for a large family of pairs (G;H) of graphs. The Ramsey number of paths was determined very early on, but surprisingly very little is known about the Ramsey number for the powers of paths. The r-th power Pr n of a path on n vertices is obtained by joining any two vertices with distance at most r. We determine the exact value of R(P2 n; P2 n) for n large and discuss some related questions
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