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Dense uniform hypergraphs have high list chromatic number and Alexandr Kostochka

Summary: Dense uniform hypergraphs have high list chromatic number
Noga Alon
and Alexandr Kostochka
October 29, 2010
Mathematics Subject Classification: 05C15, 05C35.
Keywords: List coloring, hypergraph, co-degree
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.
1 Introduction
A list for a hypergraph G is an assignment L that provides a subset Lv of a set S (called the set
of colors) to every vertex v of G. A list L for a hypergraph G is an s-list if |L(v)| = s for every
v V (G). Given a list L for G, an L-coloring of G is a proper (that is, with no monochromatic
edges) coloring f of the vertices of G such that f(v) Lv for every v V (G). The list chromatic
number (or choice number) (G) of a hypergraph G is the minimum integer s such that for
every s-list L for G, there exists an L-coloring of G. These notions were introduced (for graphs)
independently by Vizing in [10] and by Erdos, Rubin and Taylor in [5]. It turned out that list


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics