Maximum cuts and judicious partitions in graphs without short cycles

  • Authors:

  • Noga Alon;Béla Bollobás;Michael Krivelevich;Benny Sudakov

  • Affiliations:

  • Institute for Advanced Study, Princeton, NJ and Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel;Department of Mathematical Sciences, University of Memphis, Memphis TN and Trinity College, Cambridge CB2 1TQ, UK;Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel;Department of Mathematics, Princeton University, Princeton, NJ and Institute for Advanced Study, Princeton, NJ

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2003

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Abstract

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 /m2; + c(r)mr/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 Erdös. (The result and its optimality for another special case, r = 4, were already known.) 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 /m2; + δ, then there exists a partition V = V1 ∪ V2 such that both parts V1, V2 span at most /m4; - (1 - o(1))/δ2; + O(√m) edges for the case δ = o(m), and at most (1/4-Ω(1))m edges for δ = Ω(m). This enables one to extend results for the bipartite cut problem to the corresponding ones for judicious partitioning.