On a bipartition problem of Bollobás and Scott


The bipartite density of a graph G is max {|E(H)|/|E(G)|: H is a bipartite subgraph of G}. It is NP-hard to determine the bipartite density of any triangle-free cubic graph. A biased maximum bipartite subgraph of a graph G is a bipartite subgraph of G with the maximum number of edges such that one of its partite sets is independent in G. Let \( \mathcal{H} \) denote the collection of all connected cubic graphs which have bipartite density \( \tfrac{4} {5} \) and contain biased maximum bipartite subgraphs. Bollobás and Scott asked which cubic graphs belong to \( \mathcal{H} \). This same problem was also proposed by Malle in 1982. We show that any graph in \( \mathcal{H} \) can be reduced, through a sequence of three types of operations, to a member of a well characterized class. As a consequence, we give an algorithm that decides whether a given graph G belongs to \( \mathcal{H} \). Our algorithm runs in polynomial time, provided that G has a constant number of triangles that are not blocks of G and do not share edges with any other triangles in G.

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Corresponding author

Correspondence to Baogang Xu.

Additional information

Supported by NSFC Project 10671095, and Georgia Institute of Technology.

Partially supported by NSF, NSA, and NSFC Project 10628102.

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Xu, B., Yu, X. On a bipartition problem of Bollobás and Scott. Combinatorica 29, 595–618 (2009). https://doi.org/10.1007/s00493-009-2381-x

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Mathematics Subject Classification (2000)

  • 05C35
  • 05C75
  • 05C85