TY - JOUR
T1 - A quantitative Lovász criterion for Property B
AU - Ferber, Asaf
AU - Shapira, Asaf
N1 - Publisher Copyright:
© 2020 Cambridge University Press. All rights reserved.
PY - 2020/11
Y1 - 2020/11
N2 - A well-known observation of Lovász is that if a hypergraph is not 2-colourable, then at least one pair of its edges intersect at a single vertex. In this short paper we consider the quantitative version of Lovász's criterion. That is, we ask how many pairs of edges intersecting at a single vertex should belong to a non-2-colourable n-uniform hypergraph. Our main result is an exact answer to this question, which further characterizes all the extremal hypergraphs. The proof combines Bollobás's two families theorem with Pluhar's randomized colouring algorithm.
AB - A well-known observation of Lovász is that if a hypergraph is not 2-colourable, then at least one pair of its edges intersect at a single vertex. In this short paper we consider the quantitative version of Lovász's criterion. That is, we ask how many pairs of edges intersecting at a single vertex should belong to a non-2-colourable n-uniform hypergraph. Our main result is an exact answer to this question, which further characterizes all the extremal hypergraphs. The proof combines Bollobás's two families theorem with Pluhar's randomized colouring algorithm.
UR - http://www.scopus.com/inward/record.url?scp=85089826330&partnerID=8YFLogxK
U2 - 10.1017/S0963548320000334
DO - 10.1017/S0963548320000334
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AN - SCOPUS:85089826330
SN - 0963-5483
VL - 29
SP - 956
EP - 960
JO - Combinatorics Probability and Computing
JF - Combinatorics Probability and Computing
IS - 6
ER -