6 research outputs found
Degrees in oriented hypergraphs and Ramsey p-chromatic number
The family D(k,m) of graphs having an orientation such that for every vertex
v ∈ V (G) either (outdegree) deg+(v) ≤ k or (indegree) deg−(v) ≤ m have been investigated
recently in several papers because of the role D(k,m) plays in the efforts to
estimate the maximum directed cut in digraphs and the minimum cover of digraphs
by directed cuts. Results concerning the chromatic number of graphs in the family
D(k,m) have been obtained via the notion of d-degeneracy of graphs. In this paper we
consider a far reaching generalization of the family D(k,m), in a complementary form,
into the context of r-uniform hypergraphs, using a generalization of Hakimi’s theorem
to r-uniform hypergraphs and by showingPeer ReviewedPostprint (published version
Degrees in oriented hypergraphs and sparse Ramsey theory
Let be an -uniform hypergraph. When is it possible to orient the edges
of in such a way that every -set of vertices has some -degree equal
to ? (The -degrees generalise for sets of vertices what in-degree and
out-degree are for single vertices in directed graphs.) Caro and Hansberg asked
if the obvious Hall-type necessary condition is also sufficient.
Our main aim is to show that this is true for large (for given ), but
false in general. Our counterexample is based on a new technique in sparse
Ramsey theory that may be of independent interest.Comment: 20 pages, 3 figure
Degrees in oriented hypergraphs and Ramsey p-chromatic number
The family D(k,m) of graphs having an orientation such that for every vertex
v ∈ V (G) either (outdegree) deg+(v) ≤ k or (indegree) deg−(v) ≤ m have been investigated
recently in several papers because of the role D(k,m) plays in the efforts to
estimate the maximum directed cut in digraphs and the minimum cover of digraphs
by directed cuts. Results concerning the chromatic number of graphs in the family
D(k,m) have been obtained via the notion of d-degeneracy of graphs. In this paper we
consider a far reaching generalization of the family D(k,m), in a complementary form,
into the context of r-uniform hypergraphs, using a generalization of Hakimi’s theorem
to r-uniform hypergraphs and by showingPeer Reviewe
Degrees in oriented hypergraphs and Ramsey p-chromatic number
The family D(k,m) of graphs having an orientation such that for every vertex
v ∈ V (G) either (outdegree) deg+(v) ≤ k or (indegree) deg−(v) ≤ m have been investigated
recently in several papers because of the role D(k,m) plays in the efforts to
estimate the maximum directed cut in digraphs and the minimum cover of digraphs
by directed cuts. Results concerning the chromatic number of graphs in the family
D(k,m) have been obtained via the notion of d-degeneracy of graphs. In this paper we
consider a far reaching generalization of the family D(k,m), in a complementary form,
into the context of r-uniform hypergraphs, using a generalization of Hakimi’s theorem
to r-uniform hypergraphs and by showingPeer Reviewe
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Tilings and other combinatorial results
In this dissertation we treat three tiling problems and three problems in combinatorial geometry, extremal graph theory and sparse Ramsey theory.
We first consider tilings of . In this setting a tile is just a finite subset of . We say that tiles if the latter set admits a partition into isometric copies of . Chalcraft observed that there exist that do not tile but tile for some . He conjectured that such exists for any given tile. We prove this conjecture in Chapter 2.
In Chapter 3 we prove a conjecture of Lonc, stating that for any poset of size a power of , if has a greatest and a least element, then there is a positive integer such that can be partitioned into copies of .
The third tiling problem is about vertex-partitions of the hypercube graph . Offner asked: if is a subgraph of such is a power of , must , for some , admit a partition into isomorphic copies of ? In Chapter 4 we answer this question in the affirmative.
We follow up with a question in combinatorial geometry. A line in a planar set is a maximal collinear subset of . P\'or and Wood considered colourings of finite without large lines with a bounded number of colours. In particular, they examined whether monochromatic lines always appear in such colourings provided that is large. They conjectured that for all there exists an such that if and does not contain a line of cardinality larger than , then every colouring of with colours produces a monochromatic line. In Chapter 5 we construct arbitrarily large counterexamples for the case .
We follow up with a problem in extremal graph theory. For any graph, we say that a given edge is triangular if it forms a triangle with two other edges. How few triangular edges can there be in a graph with vertices and edges? For sufficiently large we prove a conjecture of F\"uredi and Maleki that gives an exact formula for this minimum. This proof is given in Chapter 6.
Finally, Chapter 7 is concerned with degrees of vertices in directed hypergraphs. One way to prescribe an orientation to an -uniform graph is to assign for each of its edges one of the possible orderings of its elements. Then, for any -set of vertices and any -set of indices , we define the -degree of to be the number of edges containing vertices in precisely the positions labelled by . Caro and Hansberg were interested in determining whether a given -uniform hypergraph admits an orientation where every set of vertices has some -degree equal to . They conjectured that a certain Hall-type condition is sufficient. We show that this is true for large, but false in general.EPSR