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Maximum cuts and judicious partitions in graphs without short Bela Bollobas

Summary: Maximum cuts and judicious partitions in graphs without short
Noga Alon
B´ela Bollob´as
Michael Krivelevich
Benny Sudakov §
We consider the bipartite cut and the judicious partition problems in graphs of girth at least
4. For the bipartite cut problem we show that every graph G with m edges, whose shortest cycle
has length at least r 4, has a bipartite subgraph with at least m
2 + c(r)m
r+1 edges. The order
of the error term in this result is shown to be optimal for r = 5 thus settling a special case of
a conjecture of Erdos. (The result and its optimality for another special case, r = 4, have been
known before.) For judicious partitions, we prove a general result as follows: if a graph G = (V, E)
with m edges has a bipartite cut of size m
2 + , then there exists a partition V = V1 V2 such
that both parts V1, V2 span at most m
4 - (1 - o(1))


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


Collections: Mathematics