The MAX-LABEL-COVER(K,L) Problem: An instance to the MAX-LABEL-COVER(K,L) problem comprises of a bipartite graph G(U, V, E) with U and V being the two partitions, and a projection function πv→u : L → K on each edge (u, v). The objective is to obtain a labeling of the vertices (a function f : U → K, V → L) which satisfies maximum fraction of the edges. Edge (u, v) is satisfied by labeling f if a...

Given that we want a 1-bit query data structure, the querying algorithm basically suggests itself. Consider a bipartite graph G, whose pieces L and R stand for the Universe and Memory bits, respectively (so |L| = m). Based on the input set S ⊆ L, for every vertex of R we decide whether to write either 0 or 1 on it. Then given a query x ∈ L, we pick a random neighbor y of x and output “x ∈ S” if...

The line graph of G, denoted L(G), is the graph with vertex set E(G), where vertices x and y are adjacent in L(G) iff edges x and y share a common vertex in G. In this paper, we determine all graphs G for which L(G) is a circulant graph. We will prove that if L(G) is a circulant, then G must be one of three graphs: the complete graph K4, the cycle Cn, or the complete bipartite graph Ka,b, for s...

Let G be a graph with vertex set V(G) and edge set E(G), respectively. The set of vertices adjacent to an x e V(G) is denoted by F(x), and the degree of x is d(x) = I F(x)l . For any subset V 'g V(G), let G[ V] denote the subgraph of G induced by the vertices of V' . Further, let K n stand for the complete graph on n vertices. It is easily seen (e.g ., Erdös [7]) that every graph G with n verti...

Abstract In this paper, we show that Leavitt path algebras of weighted graphs and separated are intimately related. We prove any algebra $L(E,\omega )$ a row-finite vertex graph $(E,\omega is $*$ -isomorphic to the lower certain bipartite $(E(\omega ),C(\omega ))$ . For general locally finite $(E, \omega , quotient $L_1(E,\omega an upper another $(E(w)_1,C(w)^1)$ furthermore introduce ${L^{\mat...

In this paper, we consider the LOCC distinguishability of product states. We employ polygons to analyze orthogonal states in any system show that with protocols, distinguish seven states, one can exclude four via a single copy. bipartite systems, result implies N are distinguishable if $$\left\lceil \frac{N}{4}\right\rceil $$ copies allowed, where l\right\rceil for real number l means smallest ...

The problem of minimum color sum of a graph is to color the vertices of the graph such that the sum (average) of all assigned colors is minimum. Recently, in [BBH+96], it was shown that in general graphs this problem cannot be approximated within n1 , for any > 0, unless NP = ZPP . In the same paper, a 9=8-approximation algorithm was presented for bipartite graphs. The hardness question for thi...

In this paper we shall consider ordinary graphs, that is, graphs without loops and multiple edges . Given a graph L, ex(n , L) will denote the maximum number of edges a graph G" of order n can have without containing any L . Determining ex(n,L), or at least finding good bounds on it will be called TURÁN TYPE EXTREMAL PROBLEM. Assume that a graph G" has E > ex(n , L) edges. Then it must contain ...