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Register a new userThe competition hypergraphs of doubly partial orders
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Since Cho and Kim (2005) showed that the competition graph of a doubly partial order is an interval graph, it has been actively studied whether or not the same phenomenon occurs for other variants of competition graph and interesting results have been obtained. Continuing in the same spirit, we study the competition hypergraph, an interesting variant of the competition graph, of a doubly partial order. Though it turns out that the competition hypergraph of a doubly partial order is not always interval, we completely characterize the competition hypergraphs of doubly partial orders which are interval.
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We define toric partial orders, corresponding to regions of graphic toric hyperplane arrangements, just as ordinary partial orders correspond to regions of graphic hyperplane arrangements. Combinatorially, toric posets correspond to finite posets under the equivalence relation generated by converting minimal elements into maximal elements, or sources into sinks. We derive toric analogues for several features of ordinary partial orders, such as chains, antichains, transitivity, Hasse diagrams, linear extensions, and total orders.
Let $mathcal{G}$ be a $k$-uniform hypergraph, $mathcal{L}_{mathcal{G}}$ be its Laplacian tensor. And $beta( mathcal{G})$ denotes the maximum number of linearly independent nonnegative eigenvectors of $mathcal{L}_{mathcal{G}}$ corresponding to the eigenvalue $0$. In this paper, $beta( mathcal{G})$ is called the geometry connectivity of $mathcal{G}$. We show that the number of connected components of $mathcal{G}$ equals the geometry connectivity $beta( mathcal{G})$.
Frankl and Furedi (1989) conjectured that the $r$-graph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${mathbb N}^{(r)}$ has the largest graph-Lagrangian of all $r$-graphs with $m$ edges. In this paper, we establish some bounds for graph-Lagrangians of some special $r$-graphs that support this conjecture.
A 2-coloring of a hypergraph is a mapping from its vertices to a set of two colors such that no edge is monochromatic. Let $H_k(n,m)$ be a random $k$-uniform hypergraph on $n$ vertices formed by picking $m$ edges uniformly, independently and with replacement. It is easy to show that if $r geq r_c = 2^{k-1} ln 2 - (ln 2) /2$, then with high probability $H_k(n,m=rn)$ is not 2-colorable. We complement this observation by proving that if $r leq r_c - 1$ then with high probability $H_k(n,m=rn)$ is 2-colorable.
An oriented hypergraph is an oriented incidence structure that extends the concepts of signed graphs, balanced hypergraphs, and balanced matrices. We introduce hypergraphic structures and techniques that generalize the circuit classification of the signed graphic frame matroid to any oriented hypergraphic incidence matrix via its locally-signed-graphic substructure. To achieve this, Camions algorithm is applied to oriented hypergraphs to provide a generalization of reorientation sets and frustration that is only well-defined on balanceable oriented hypergraphs. A simple partial characterization of unbalanceable circuits extends the applications to representable matroids demonstrating that the difference between the Fano and non-Fano matroids is one of balance.
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