Asymptotic properties of random regular graphs are object of extensive study in mathematics. In this note we argue, based on theory of spin glasses, that in random regular graphs the maximum cut size asymptotically equals the number of edges in the graph minus the minimum bisection size. Maximum cut and minimal bisection are two famous NP-complete problems with no known general relation between...

We give O(log n)-approximation algorithm based on the cut-matching framework of [10, 13, 14] for the computing the sparsest cut on directed graphs. Our algorithm uses only O(log n) single commodity max-flow computations and thus breaks the multicommodity-flow barrier for computing the sparsest cut on directed graphs.

1.1. Preamble. Given a graph G = V E and a subset A ⊆ V , the cut determined by A is the vector A ∈ ±1 E with ijth entry −1 if and only if A∩ i j = 1. The cut polytope CUT G is the polytope in E defined as the convex hull of all cuts A (A⊆ V ). Given edge weights w ∈ E , the max-cut problem is the problem of finding a cut A whose weight ∑ ij∈E i∈A j ∈A wij is maximum. Hence, it can be formulate...

We investigate a graph function which is related to the local density, the maximal cut and the least eigenvalue of a graph. In particular it enables us to prove the following assertions: Let p 3 be integer, c 2 (0; 1=2) and G be a Kp-free graph on n vertices with e cn2 edges. There exists a positive constant = (c; p) such that a) some bn=2c subset of V (G) induces at most c 4 n edges (this answ...

This is a summary of the proof by G.E. Coxson [1] that P-matrix recognition is co-NP-complete. The result follows by a reduction from the MAX CUT problem using results of S. Poljak and J. Rohn [5].

We extend the colourful complete bipartite subgraph theorems of [G. Simonyi, G. Tardos, Local chromatic number, Ky Fan’s theorem, and circular colorings, Combinatorica 26 (2006), 587–626] and [G. Simonyi, G. Tardos, Colorful subgraphs of Kneser-like graphs, European J. Combin. 28 (2007), 2188–2200] to more general topological settings. We give examples showing that the hypotheses are indeed mor...

Throughout the paper, G denotes a graph that has no loops, but that may have multiple edges. The multi-set of edges of G is denoted by E(G). A graph G is simple if it has no (loops or) multiple edges. A biclique of G is a simple complete bipartite subgraph of G. A biclique decomposition of G is a collection of bicliques of G, such that each edge of G is in precisely one of the bicliques in the ...

In a 1965 paper, Erdős remarked that a graph G has a bipartite subgraph that has at least half the number of edges of G. The purpose of this note is to prove a matroid analogue of Erdős’s original observation. It follows from this matroid result that every loopless binary matroid has a restriction that uses more than half of its elements and has no odd circuits; and, for 2 ≤ k ≤ 5, every bridge...

A separation algorithm is a procedure for generating cutting planes. Up to now, only a few polynomial-time separation algorithms were known for the Boolean quadric and cut polytopes. These polytopes arise in connection with zero-one quadratic programming and the maxcut problem, respectively. We present a new algorithm, which separates over a class of valid inequalities that includes all odd bic...

Definition 1.1. Given a (undirected) graph G = (V,E), the maximum cut δ(U) for U ⊆ V is the cut with maximal value |δ(U)|. The Maximum Cut Problem consists of finding a maximal cut. We let MaxCut(G) = max{|δ(U)| : U ⊆ V } be the value of the maximum cut in G, and MaxCut′ = MaxCut(G) |E| be the normalized version (note that both normalized and unnormalized maximum cut values are induced by the s...