On a bottleneck bipartition conjecture of Erdős

Abstract

For a graphG, let γ(U,V)=max{e(U), e(V)} for a bipartition (U, V) ofV(G) withUυV=V(G),UφV=Ø. Define γ(G)=min(U,V ){γ(U,V)}. Paul Erdős conjectures\(\gamma (G)/e(G) \leqslant {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4} + O\left( {1/\sqrt {e(G)} } \right)\). This paper verifies the conjecture and shows\(\gamma (G)/e(G) \leqslant {1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}\left( {1 + \sqrt {2/e(G)} } \right)\).

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References

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    L. Clark, F. Shahrokhi, andL. A. Székely: A Lineartime Algorithm For Graph Partition Problems, to appear inInform. Proc. Letters.

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    R. Entringer: Personal communication.

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    P. Erdős: Personal communication.

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This work was part of the author's Ph. D. thesis at the University of New Mexico. Research Partially supported by NSA Grant MDA904-92-H-3050.

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Porter, T.D. On a bottleneck bipartition conjecture of Erdős. Combinatorica 12, 317–321 (1992). https://doi.org/10.1007/BF01285820

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AMS Subject Classification code (1991)

  • 05 C 35