A rainbow matching in an edge-colored graph is a matching whose edges have distinct colors. We address the complexity issue of the following problem, max rainbow matching: Given an edge-colored graph G, how large is the largest rainbow matching in G? We present several sharp contrasts in the complexity of this problem. We show, among others, that • max rainbow matching can be approximated by a ...

The mean value of the matching polynomial is computed in the family of all labeled graphs with n vertices. We introduce the dominating polynomial of a graph whose coefficients enumerate the dominating sets for a graph and study some properties of the polynomial. The mean value of this polynomial is determined in a certain special family of bipartite digraphs.

In recent years, several powerful techniques have been developed to design randomized polynomial-space parameterized algorithms. In this paper, we introduce an enhancement of color coding to design deterministic polynomial-space parameterized algorithms. Our approach aims at reducing the number of random choices by exploiting the special structure of a solution. Using our approach, we derive po...

An induced matching M in a graph G is dominating if every edge not in M shares exactly one vertex with an edge in M . The dominating induced matching problem (also known as efficient edge domination) asks whether a graph G contains a dominating induced matching. This problem is generally NP-complete, but polynomial-time solvable for graphs with some special properties. In particular, it is solv...

Given a graphG = (V,E) and for each vertex v ∈ V a subsetB(v) of the set {0, 1, . . . , dG(v)}, where dG(v) denotes the degree of vertex v in the graph G, a B-matching of G is any set F ⊆ E such that dF (v) ∈ B(v) for each vertex v, where dF (v) denotes the number of edges of F incident to v. The general matching problem asks the existence of aB-matching in a given graph. A setB(v) is said to h...

The Geometry of Polynomials, also known as the analytic theory of polynomials, refers the study of the zero loci of polynomials with complex coefficients (and their dynamics under various transformations of the polynomials) using methods of real and complex analysis. The course will focus on the fragment of this subject which deals with real-rooted polynomials and their multivariate generalizat...

Let G be a simple graph and let S(G) be the subdivision graph of G, which is obtained from G by replacing each edge of G by a path of length two. In this paper, by the Principle of Inclusion and Exclusion we express the matching polynomial and Hosoya index of S(G) in terms of the matchings of G. Particularly, if G is a regular graph or a semi-regular bipartite graph, then the closed formulae of...

We study the problem of approximating value matching polynomial on graphs with edge parameter γ, where γ takes arbitrary values in complex plane. When is a positive real, Jerrum and Sinclair showed that admits an FPRAS general graphs. For Patel Regts, building methods developed by Barvinok, FPTAS maximum degree Δ as long not negative real number less than or equal to −1/(4(Δ −1)). Our first mai...

I have computed the matching polynomials of a number isomers fullerenes various sizes with objective developing molecular descriptors and similarity measures for on basis their polynomials. Two novel polynomial-based topological are developed, they demonstrated to discriminating power contrast closely related fullerenes. The ways place up seven disjoint dimers fullerene shown be identical, as n...