Large and Judicious Bisections of Graphs
نویسنده
چکیده
It is very well known that every graph on n vertices and m edges admits a bipartition of size at least m/2. This bound can be improved to m/2 + (n − 1)/4 for connected graphs, and m/2 + n/6 for graphs without isolated vertices, as proved by Edwards, and Erdős, Gyárfás, and Kohayakawa, respectively. A bisection of a graph is a bipartition in which the size of the two parts differ by at most 1. We prove that graphs with maximum degree o(n) in fact contain a bisection which asymptotically achieves the above bounds. All these results follow from a more general theorem, which can also be used to prove several conjectures of Bollobás and Scott on bisections of graphs. Joint work with Po-Shen Loh and Benny Sudakov.
منابع مشابه
Bisections of graphs
A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several questions and conjectures of Bollobás and Scott, we study maximum bisections of graphs. First, we extend the classical Edwards bound on maximum cuts to bisections. A ...
متن کاملOn judicious bisections of graphs
A bisection of a graph G is a bipartition S1, S2 of V (G) such that −1 ≤ |S1|− |S2| ≤ 1. It is NP-hard to find a bisection S1, S2 of a graph G maximizing e(S1, S2) (respectively, minimizing max{e(S1), e(S2)}), where e(S1, S2) denotes the number of edges of G between S1 and S2, and e(Si) denotes the number of edges of G with both ends in Si. There has been algorithmic work on bisections, but ver...
متن کاملGraph Bisection AlgoriLhrns With Good A-veragc Case Behavior
In the paper, we describe a polynomial time algorithm that, for every input graph, either outputs the minimum bisection of the graph or halts without output. More importantly, we show that the algorithm chooses the former course with high probability for many natural classes of graphs. In particular, for every fixed d 2 3, all suffciently large n and all b = o(nl l / lT j ) , the algorithm find...
متن کاملGraph bisection algorithms
In this thesis, we describe a polynomial time algorithm that, for every input graph, either outputs the minimum bisection of the graph or halts without output. More importantly, we show that the algorithm chooses the former course with high probability for many natural classes of graphs. In particular, for every fixed d > 3, all sufficiently large n and all b = o(ni-1/d2lJ), the algorithm finds...
متن کاملOn the Maximal Error of Spectral Approximation of Graph Bisection
Spectral graph bisections are a popular heuristic aimed at approximating the solution of the NP-complete graph bisection problem. This technique, however, does not always provide a robust tool for graph partitioning. Using a special class of graphs, we prove that the standard spectral graph bisection can produce bisections that are far from optimal. In particular, we show that the maximum error...
متن کامل