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MaxCut in H-free graphs Michael Krivelevich

Summary: MaxCut in H-free graphs
Noga Alon
Michael Krivelevich
Benny Sudakov
Dedicated to B´ela Bollob´as on his 60th birthday
For a graph G, let f(G) denote the maximum number of edges in a cut of G. For an integer
m and for a fixed graph H, let f(m, H) denote the minimum possible cardinality of f(G),
as G ranges over all graphs on m edges that contain no copy of H. In this paper we study
this function for various graphs H. In particular we show that for any graph H obtained by
connecting a single vertex to all vertices of a fixed nontrivial forest, there is a c(H) > 0 such
that f(m, H) m
2 + c(H)m4/5
, and this is tight up to the value of c(H). We also prove that
for any even cycle C2k there is a c(k) > 0 such that f(m, C2k) m
2 + c(k)m(2k+1)/(2k+2)
this is tight, up to the value of c(k), for 2k {4, 6, 10}. The proofs combine combinatorial,
probabilistic and spectral techniques.
1 Introduction


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


Collections: Mathematics