43 research outputs found
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
Rainbow spanning subgraphs in bounded edge–colourings of graphs with large minimum degree
We study the existence of rainbow perfect matching and rainbow Hamiltonian cycles in edge–colored graphs where every color appears a bounded number of times. We derive asymptotically tight bounds on the minimum degree of the host graph for the existence of such rainbow spanning structures. The proof uses a probabilisitic argument combined with switching techniques.Postprint (updated version
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
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Matchings and Covers in Hypergraphs
In this thesis, we study three variations of matching and covering problems in hypergraphs. The first is motivated by an old conjecture of Ryser which says that if \mcH is an -uniform, -partite hypergraph which does not have a matching of size at least , then \mcH has a vertex cover of size at most . In particular, we examine the extremal hypergraphs for the case of Ryser's conjecture. In 2014, Haxell, Narins, and Szab{\'{o}} characterized these -uniform, tripartite hypergraphs. Their work relies heavily on topological arguments and seems difficult to generalize. We reprove their characterization and significantly reduce the topological dependencies. Our proof starts by using topology to show that every -uniform, tripartite hypergraph has two matchings which interact with each other in a very restricted way. However, the remainder of the proof uses only elementary methods to show how the extremal hypergraphs are built around these two matchings.
Our second motivational pillar is Tuza's conjecture from 1984. For graphs and , let denote the size of a maximum collection of pairwise edge-disjoint copies of in and let denote the minimum size of a set of edges which meets every copy of in . The conjecture is relevant to the case where and says that for every graph . In 1998, Haxell and Kohayakawa proved that if is a tripartite graph, then . We use similar techniques plus a topological result to show that for all tripartite graphs . We also examine a special subclass of tripartite graphs and use a simple network flow argument to prove that for all such graphs .
We then look at the problem of packing and covering edge-disjoint 's. Yuster proved that if a graph does not have a fractional packing of 's of size bigger than , then . We give a complementary result to Yuster's for 's: We show that every graph has a fractional cover of 's of size at most . We also provide upper bounds on for several classes of graphs.
Our final topic is a discussion of fractional stable matchings. Tan proved that every graph has a -integral stable matching. We consider hypergraphs. There is a natural notion of fractional stable matching for hypergraphs, and we may ask whether an analogous result exists for this setting. We show this is not the case: Using a construction of Chung, F{\"{u}}redi, Garey, and Graham, we prove that, for all n \in \mbN, there is a -uniform hypergraph with preferences with a fractional stable matching that is unique and has denominators of size at least