Maximum edge-cuts in cubic graphs with large girth and in random cubic graphs

Authors:
František Kardoš;Daniel Král′;Jan Volec
Affiliations:
Institute of Mathematics, Faculty of Science, University of Pavol Jozef Šafárik, 041 54 Košice, Slovakia;Institute for Theoretical Computer Science, Charles University, Prague, Czech Republic and Computer Science Institute, Faculty of Mathematics and Physics, Charles University, 11800 Prague, Czech R ...;Computer Science Institute, Faculty of Mathematics and Physics, Charles University, 11800 Prague, Czech Republic
Venue:
Random Structures & Algorithms
Year:
2012

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Abstract

We show that for every cubic graph Gwith sufficiently large girth there exists a probability distribution on edge-cuts in Gsuch that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that Gcontains an edge-cut of size at least 1.33008n, where nis the number of vertices of G, and has fractional cut covering number at most 1.127752. The lower bound on the size of maximum edge-cut also applies to random cubic graphs. Specifically, a random n-vertex cubic graph a.a.s. contains an edge-cut of size 1.33008n- o(n). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012 © 2012 Wiley Periodicals, Inc.