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Search results for: HYPERGRAPH
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Equitable coloring of hypergraphs
PublicationA hypergraph is equitablyk-colorable if its vertices can be partitioned into k sets/colorclasses in such a way that monochromatic edges are avoided and the number of verticesin any two color classes differs by at most one. We prove that the problem of equitable 2-coloring of hypergraphs is NP-complete even for 3-uniform hyperstars. Finally, we apply the method of dynamic programming for designing a polynomial-time algorithm to...
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Weighted 2-sections and hypergraph reconstruction
PublicationIn the paper we introduce the notion of weighted 2-sections of hypergraphs with integer weights and study the following hypergraph reconstruction problems: (1) Given a weighted graph , is there a hypergraph H such that is its weighted 2-section? (2) Given a weighted 2-section , find a hypergraph H such that is its weighted 2-section. We show that (1) is NP-hard even if G is a complete graph or integer weights w does not exceed...
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Edge-coloring of 3-uniform hypergraphs
PublicationWe consider edge-colorings of 3-uniform hypergraphs which is a natural generalization of the problem of edge-colorings of graphs. Various classes of hypergraphs are discussed and we make some initial steps to establish the border between polynomial and NP-complete cases. Unfortunately, the problem appears to be computationally difficult even for relatively simple classes of hypergraphs.
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Independence in uniform linear triangle-free hypergraphs
PublicationThe independence number a(H) of a hypergraph H is the maximum cardinality of a set of vertices of H that does not contain an edge of H. Generalizing Shearer’s classical lower bound on the independence number of triangle-free graphs Shearer (1991), and considerably improving recent results of Li and Zang (2006) and Chishti et al. (2014), we show a new lower bound for a(H) for an r-uniform linear triangle-free hypergraph H with r>=2.
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Towards the boundary between easy and hard control problems in multicast Clos networks
PublicationIn this article we study 3-stage Clos networks with multicast calls in general and 2-cast calls, in particular. We investigate various sizes of input and output switches and discuss some routing problems involved in blocking states. To express our results in a formal way we introduce a model of hypergraph edge-coloring. A new class of bipartite hypergraphs corresponding to Clos networks is studied. We identify some polynomially...
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On Tradeoffs Between Width- and Fill-like Graph Parameters
PublicationIn this work we consider two two-criteria optimization problems: given an input graph, the goal is to find its interval (or chordal) supergraph that minimizes the number of edges and its clique number simultaneously. For the interval supergraph, the problem can be restated as simultaneous minimization of the path width pw(G) and the profile p(G) of the input graph G. We prove that for an arbitrary graph G and an integer t ∈ {1,...
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A bound on the number of middle-stage crossbars in f-cast rearrangeable Clos networks
PublicationIn 2006 Chen and Hwang gave a necessary and sufficient condition under which a three-stage Clos network is rearrangeable for broadcast connections. Assuming that only crossbars of the first stage have no fan-out property, we give similar conditions for f-cast Clos networks, where f is an arbitrary but fixed invariant of the network. Such assumptions are valid for some practical switching systems, e.g. high-speed crossconnects....
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The Potential of Greed for Independence
PublicationThe well-known lower bound on the independence number of a graph due to Caro and Wei can be established as a performance guarantee of two natural and simple greedy algorithms or of a simple randomized algorithm. We study possible generalizations and improvements of these approaches using vertex weights and discuss conditions on so-called potential functions p(G) : V(G) -> N_0 defined on the vertex set of a graph G for which suitably...