TY - JOUR

T1 - Choosability in random hypergraphs

AU - Krivelevich, Michael

AU - Vu, Van H.

N1 - Funding Information:
1Research was partially performed while this author was with DIMACS Center, Rutgers University, Piscataway, NJ 08854, USA. His research was supported by a DIMACS Postdoctoral Fellowship, by a USA Israeli BSF Grant, and by A. Bergmann Memorial Grant.

PY - 2001

Y1 - 2001

N2 - The choice number of a hypergraph H=(V, E) is the least integer s for which, for every family of color lists σl={S(v): v ∈ V}, satisfying |S(v)| = s for every v ∈ V, there exists a choice function f so that f(v)∈S(v) for every v∈V, and no edge of H is monochromatic under f. In this paper we consider the asymptotic behavior of the choice number of a random k-uniform hypergraph H(k, n, p). Our main result states that for every k≥2 and for all values of the edge probability p = p(n) down to p=O(n-k+1) the ratio between the choice number and the chromatic number of H(k, n, p) does not exceed k1/(k-1) asymptotically. Moreover, for large values of p, namely, when p≥n-(k-1)2/(2k)+for an arbitrary positive constant , the choice number and the chromatic number of H(k, n, p) have almost surely the same asymptotic value.

AB - The choice number of a hypergraph H=(V, E) is the least integer s for which, for every family of color lists σl={S(v): v ∈ V}, satisfying |S(v)| = s for every v ∈ V, there exists a choice function f so that f(v)∈S(v) for every v∈V, and no edge of H is monochromatic under f. In this paper we consider the asymptotic behavior of the choice number of a random k-uniform hypergraph H(k, n, p). Our main result states that for every k≥2 and for all values of the edge probability p = p(n) down to p=O(n-k+1) the ratio between the choice number and the chromatic number of H(k, n, p) does not exceed k1/(k-1) asymptotically. Moreover, for large values of p, namely, when p≥n-(k-1)2/(2k)+for an arbitrary positive constant , the choice number and the chromatic number of H(k, n, p) have almost surely the same asymptotic value.

UR - http://www.scopus.com/inward/record.url?scp=0035200859&partnerID=8YFLogxK

U2 - 10.1006/jctb.2001.2053

DO - 10.1006/jctb.2001.2053

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AN - SCOPUS:0035200859

VL - 83

SP - 241

EP - 257

JO - Journal of Combinatorial Theory. Series B

JF - Journal of Combinatorial Theory. Series B

SN - 0095-8956

IS - 2

ER -