Edwards showed that every graph of size m ≥ 1 has a bipartite subgraph of size at least m/2 + √ m/8 + 1/64− 1/8. We show that every graph of size m ≥ 1 has a bipartition in which the Edwards bound holds, and in addition each vertex class contains at most m/4 + √ m/32 + 1/256 − 1/16 edges. This is exact for complete graphs of odd order, which we show are the only extremal graphs without isolated...

We introduce and evaluate a very simple landmark-based network partition technique called Hierarchical Bipartition Routing (HBR) to support greedy routing with delivery guarantee in wireless ad hoc sensor networks. The main assets of HBR are: 1) packet delivery guarantee, 2) unique virtual addressing, 3) no packet overhead in typical cases, 4) small routing tables, 5) no physical geographic coo...

In which we introduce the Leighton-Rao relaxation of sparsest cut. Let G = (V, E) be an undirected graph. Unlike past lectures, we will not need to assume that G is regular. We are interested in finding a sparsest cut in G, where the sparsity of a non-trivial bipartition (S, V − S) of the vertices is which is the ratio between the fraction of edges that are cut by (S, V − S) and the fraction of...

Classical to quantum decoherence transition, an issue existing for incoherent superposition of Bell-diagonal states is studied for three dimensional bipartite AB mixed quantum systems. Depending on the initial conditions, the dynamics of classical and quantum correlations can exhibit a sudden transition between classical to quantum decoherence. This result is calculated numerically by using ent...

A function f : P ! P , P = f0; 1; : : : ; p 1g is kdecomposable i f can be represented as f(X1;X2) = g(h1(X1); h2(X1); : : : ; hk(X1);X2), where (X1; X2) is a bipartition of input variables. This paper introduces the notion of totally k-undecomposable functions. By using this concept, we can drastically reduce the search space to nd k-decompositions. A systematic method to nd the bipartitions o...

Let G be a connected k–regular bipartite graph with bipartition V (G) = X ∪Y and adjacency matrix A. We say G is det–extremal if per(A) = |det(A)|. Det–extremal k–regular bipartite graphs exist only for k = 2 or 3. McCuaig has characterized the det–extremal 3–connected cubic bipartite graphs. We extend McCuaig’s result by determining the structure of det–extremal cubic bipartite graphs of conne...

Let E be a possibly infinite set and let M N matroids defined on E. We say that the pair {M,N} has Intersection property if share an independent I admitting bipartition IM⊔IN such spanM(IM)∪spanN(IN)=E. The Matroid Conjecture of Nash-Williams says every matroid property. conjecture is known easy to prove in case when one uniform it was shown by Bowler Carmesin implied its special where direct s...

Let (B,C) be an (edge-)colored bipartite graph with bipartition (X,Y ), i.e., B is assigned a mapping C : E(B) → {1, 2, · · · , r}, the set of colors. A matching of B is called heterochromatic if its any two edges have different colors. Let N (S) denote a maximum color neighborhood of S ⊆ V (B). We show that if |N (S)| ≥ |S| for all S ⊆ X, then B contains a heterochromatic matching with cardina...

The dynamical evolution of weights in the AdaBoost algorithm contains useful information about the rôle that the associated data points play in the built of the AdaBoost model. In particular, the dynamics induces a bipartition of the data set into two (easy/hard) classes. Easy points are ininfluential in the making of the model, while the varying relevance of hard points can be gauged in terms ...

The maximum stable matching problem (Max-SMP) and the minimum stable matching problem (Min-SMP) have been known to be NP-hard for subcubic bipartite graphs, while Max-SMP can be solved in polynomal time for a bipartite graph G with a bipartition (X,Y ) such that degG(v) ≤ 2 for any v ∈ X. This paper shows that both Max-SMP and Min-SMP can be solved in linear time for trees. This is the first po...