Problems and results on judicious partitions

We present a few results and a larger number of questions concerning partitions of graphs or hypergraphs, where the objective is to maximize or minimize several quantities simultaneously. We consider a variety of extremal problems; many of these also have algorithmic counterparts.

[1]  Thomas Dale Porter,et al.  On a bottleneck bipartition conjecture of Erdős , 1992, Comb..

[2]  David P. Williamson,et al.  .879-approximation algorithms for MAX CUT and MAX 2SAT , 1994, STOC '94.

[3]  Noga Alon,et al.  Maximum cuts and judicious partitions in graphs without short cycles , 2003, J. Comb. Theory B.

[4]  William Staton,et al.  Extremal bipartite subgraphs of cubic triangle-free graphs , 1982, J. Graph Theory.

[5]  Kazuoki Azuma WEIGHTED SUMS OF CERTAIN DEPENDENT RANDOM VARIABLES , 1967 .

[6]  Zsolt Tuza,et al.  Bipartite Subgraphs of Triangle-Free Graphs , 1994, SIAM J. Discret. Math..

[7]  Zsolt Tuza,et al.  Maximum cuts and largest bipartite subgraphs , 1993, Combinatorial Optimization.

[8]  Paul Erdös,et al.  How to make a graph bipartite , 1987, J. Comb. Theory, Ser. B.

[9]  Alex Scott,et al.  Better bounds for Max Cut , 2002 .

[10]  C. S. Edwards Some Extremal Properties of Bipartite Subgraphs , 1973, Canadian Journal of Mathematics.

[11]  Zsolt Tuza,et al.  Maximum cuts: Improvements and local algorithmic analogues of the Edwards-Erdös inequality , 1999, Discret. Math..

[12]  P. Erdős,et al.  COLLOQUIA MATHEMATICA SOCIETATIS JÁNOS BOLYAI 4 . COMBINATORIAL THEORY AND ITS APPLICATIONS , 1969 .

[13]  Béla Bollobás,et al.  Judicious partitions of bounded‐degree graphs , 2004, J. Graph Theory.

[14]  Noga Alon,et al.  Bipartite subgraphs , 1996, Comb..

[15]  Thomas Dale Porter Minimal Partitions of a Graph , 1999, Ars Comb..

[16]  Bing Yang,et al.  Graph Partitions II , .

[17]  Saharon Shelah,et al.  A Tribute to Paul Erdős: Graphs with no unfriendly partitions , 1990 .

[18]  Béla Bollobás,et al.  Judicious Partitions of 3-uniform Hypergraphs , 2000, Eur. J. Comb..

[19]  Alexandr V. Kostochka,et al.  On an upper bound of a graph's chromatic number, depending on the graph's degree and density , 1977, J. Comb. Theory B.

[20]  W. Hoeffding Probability Inequalities for sums of Bounded Random Variables , 1963 .

[21]  Igor Rivin,et al.  On some extremal problems in graph theory , 1999 .

[22]  David P. Williamson,et al.  Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming , 1995, JACM.

[23]  E. C. Milner,et al.  Unfriendly partitions of a graph , 1990, J. Comb. Theory, Ser. B.

[24]  Colin McDiarmid,et al.  Surveys in Combinatorics, 1989: On the method of bounded differences , 1989 .

[25]  C. V. Eynden,et al.  A proof of a conjecture of Erdös , 1969 .

[26]  James B. Shearer,et al.  A Note on Bipartite Subgraphs of Triangle-Free Graphs , 1992, Random Struct. Algorithms.

[27]  P. Erdos,et al.  Problems and Results in Graph Theory and Combinatorial Analysis , 1977 .

[28]  Noga Alon,et al.  Bipartite subgraphs of integer weighted graphs , 1998, Discret. Math..

[29]  Bruce A. Reed,et al.  An extremal function for the achromatic number , 1991, Graph Structure Theory.

[30]  Béla Bollobás,et al.  Judicious partitions of graphs , 1993 .

[31]  V. Sós Surveys in Combinatorics: IRREGULARITIES OF PARTITIONS: RAMSEY THEORY, UNIFORM DISTRIBUTION , 1983 .

[32]  Mauro Dell'Amico,et al.  Annotated Bibliographies in Combinatorial Optimization , 1997 .

[33]  Claudio Bernardi On a theorem about vertex colorings of graphs , 1987, Discret. Math..

[34]  Johan Håstad,et al.  Some optimal inapproximability results , 2001, JACM.

[35]  Béla Bollobás,et al.  Exact Bounds for Judicious Partitions of Graphs , 1999, Comb..

[36]  S. C. Locke Maximum k-colorable subgraphs , 1982, J. Graph Theory.

[37]  J. Beck,et al.  Discrepancy Theory , 1996 .

[38]  S. C. Locke,et al.  Largest bipartite subgraphs in triangle-free graphs with maximum degree three , 1986 .

[39]  Béla Bollobás,et al.  Judicious Partitions of Hypergraphs , 1997, J. Comb. Theory, Ser. A.

[40]  J. H. van Lint Über die Approximation von Zahlen durch Reihen mit positiven Gliedern , 1962 .