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Note on robust critical graphs with large odd girth
Nstase, E.,Rodl, V.,Siggers, M. North-Holland Pub. Co ; Elsevier Science Ltd 2010 Discrete mathematics Vol.310 No.3
A graph G is (k+1)-critical if it is not k-colourable but G-e is k-colourable for any edge e@?E(G). In this paper we show that for any integers k>=3 and l>=5 there exists a constant c=c(k,l)>0, such that for all n@?, there exists a (k+1)-critical graph G on n vertices with n>n@? and odd girth at least @?, which can be made (k-1)-colourable only by the omission of at least cn<SUP>2</SUP> edges.
The number of B<sub>3</sub>-sets of a given cardinality
Dellamonica, D.,Kohayakawa, Y.,Lee, S.J.,Rodl, V.,Samotij, W. Academic Press 2016 Journal of combinatorial theory. Series A Vol.142 No.-
<P>A set S of integers is a B-3-set if all the sums of the form a(1) a(2)+a(3), with a(1), a(2) and a(3) epsilon S and a(1) <= a(2) <= a(3), are distinct. We obtain asymptotic bounds for the number of B-3-sets of a given cardinality contained in the interval [n] = {1,...,n}. We use these results to estimate the maximum size of a B-3-set contained in a typical (random) subset of [n] of a given cardinality. These results confirm conjectures recently put forward by the authors [On the number of B-h-sets, Combin. Probab. Comput. 25 (2016), no. 1, 108-127]. (C) 2016 Elsevier Inc. All rights reserved.</P>