The degree, size and chromatic index of a uniform hypergraph
نویسندگان
چکیده
Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the maximum degree of a vertex of H. We conjecture that for every > 0, if D is sufficiently large as a function of t, k and , then the chromatic index of H is at most (t − 1 + 1/t + )D. We prove this conjecture for the special case of intersecting hypergraphs in the following stronger form: If H is an intersecting k-uniform hypergraph in which no two edges share more than t common vertices, and D is the maximum degree of a vertex of H, where D is sufficiently large as a function of k, then H has at most (t − 1 + 1/t)D edges.
منابع مشابه
On the Degree, Size, and Chromatic Index of a Uniform Hypergraph
Let H be a k-uniform hypergraph in which no two edges share more than t common vertices, and let D denote the maximum degree of a vertex of H. We conjecture that for every > 0, if D is sufficiently large as a function of t, k and , then the chromatic index of H is at most (t − 1 + 1/t + )D. We prove this conjecture for the special case of intersecting hypergraphs in the following stronger form:...
متن کاملOn the Chromatic Index of Random Uniform Hypergraphs
Let H(n,N), where k ≥ 2, be a random hypergraph on the vertex set [n] = {1, 2, . . . , n} with N edges drawn independently with replacement from all subsets of [n] of size k. For d̄ = kN/n and any ε > 0 we show that if k = o(ln(d̄/ lnn)) and k = o(ln(n/ ln d̄)), then with probability 1−o(1) a random greedy algorithm produces a proper edge-colouring of H(n,N) with at most d̄(1+ε) colours. This yield...
متن کاملHypergraph Colouring and Degeneracy
A hypergraph is d-degenerate if every subhypergraph has a vertex of degree at most d. A greedy algorithm colours every such hypergraph with at most d+ 1 colours. We show that this bound is tight, by constructing an r-uniform d-degenerate hypergraph with chromatic number d + 1 for all r ≥ 2 and d ≥ 1. Moreover, the hypergraph is triangle-free, where a triangle in an r-uniform hypergraph consists...
متن کاملDense uniform hypergraphs have high list chromatic number
The first author showed that the list chromatic number of every graph with average degree d is at least (0.5 − o(1)) log2 d. We prove that for r ≥ 3, every r-uniform hypergraph in which at least half of the (r − 1)-vertex subsets are contained in at least d edges has list chromatic number at least ln d 100r3 . When r is fixed, this is sharp up to a constant factor.
متن کاملNear-optimal list colorings
We show that a simple variant of a naive colouring procedure can be used to list colour the edges of a k-uniform linear hypergraph of maximum degree provided every vertex has a list of at least +c(log) 4 1? 1 k available colours (where c is a constant which depends on k). We can extend this to colour hypergraphs of maximum codegree o(() with + o(() colours. This improves earlier results of Kahn...
متن کامل