6 research outputs found

    Degrees in oriented hypergraphs and Ramsey p-chromatic number

    No full text
    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

    Full text link
    Let GG be an rr-uniform hypergraph. When is it possible to orient the edges of GG in such a way that every pp-set of vertices has some pp-degree equal to 00? (The pp-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 rr large (for given pp), 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

    No full text
    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

    No full text
    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
    corecore