2,244 research outputs found
Universal Hypergraphs.
In this thesis, we study universal hypergraphs. What are these? Let us start with defining a universal graph as a graph on n vertices that contains each of the many possible graphs of a smaller size k \u3c n as an induced subgraph. A hypergraph is a discrete structure on n vertices in which edges can be of any size, unlike graphs, where the edge size is always two. If all edges are of size three, then the hypergraph is said to be 3-uniform. If a 3-uniform hypergraph can have edges colored one of a colors, then it is called a 3-uniform hypergraph with a colors. Analogously with universal graphs, a universal, induced, 3-uniform, k-hypergraph, with a possible edge colors is then defined to be a 3-uniform a-colored hypergraph on n vertices that contains each of the many possible 3-uniform a-colored hypergraphs on k vertices, k \u3c n. In this thesis, we study conditions for the existence of a such a universal hypergraph, and address the question of how large n must be, given a fixed k, so that hypergraphs on n vertices are universal with high probability. This extends the work of Alon, [2] who studied the case of a = 2, and that too for graphs (not hypergraphs)
Total Domishold Graphs: a Generalization of Threshold Graphs, with Connections to Threshold Hypergraphs
A total dominating set in a graph is a set of vertices such that every vertex
of the graph has a neighbor in the set. We introduce and study graphs that
admit non-negative real weights associated to their vertices such that a set of
vertices is a total dominating set if and only if the sum of the corresponding
weights exceeds a certain threshold. We show that these graphs, which we call
total domishold graphs, form a non-hereditary class of graphs properly
containing the classes of threshold graphs and the complements of domishold
graphs, and are closely related to threshold Boolean functions and threshold
hypergraphs. We present a polynomial time recognition algorithm of total
domishold graphs, and characterize graphs in which the above property holds in
a hereditary sense. Our characterization is obtained by studying a new family
of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of
independent interest.Comment: 19 pages, 1 figur
Bounding the Number of Hyperedges in Friendship -Hypergraphs
For , an -uniform hypergraph is called a friendship
-hypergraph if every set of vertices has a unique 'friend' - that
is, there exists a unique vertex with the property that for each
subset of size , the set is a hyperedge.
We show that for , the number of hyperedges in a friendship
-hypergraph is at least , and we
characterise those hypergraphs which achieve this bound. This generalises a
result given by Li and van Rees in the case when .
We also obtain a new upper bound on the number of hyperedges in a friendship
-hypergraph, which improves on a known bound given by Li, van Rees, Seo and
Singhi when .Comment: 14 page
Natural realizations of sparsity matroids
A hypergraph G with n vertices and m hyperedges with d endpoints each is
(k,l)-sparse if for all sub-hypergraphs G' on n' vertices and m' edges, m'\le
kn'-l. For integers k and l satisfying 0\le l\le dk-1, this is known to be a
linearly representable matroidal family.
Motivated by problems in rigidity theory, we give a new linear representation
theorem for the (k,l)-sparse hypergraphs that is natural; i.e., the
representing matrix captures the vertex-edge incidence structure of the
underlying hypergraph G.Comment: Corrected some typos from the previous version; to appear in Ars
Mathematica Contemporane
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