Let G be a simple graph having a maximum matching M. The deficiency def(G) of G is the number of M-unsaturated vertices in G. A problem that arises is that of determining the set of possible values of def(G). In this paper we present a solution for the case of r-regular k-edge-connected graphs. 1. INTRODUCTION In this paper the graphs are finite, loopless and have no multiple edges. For the mos...

Given a graph G = (V,E) on n vertices, the MAXIMUM r-REGULAR INDUCED CONNECTED SUBGRAPH (r-MaxRICS) problems ask for a maximum sized subset of vertices S ⊆ V such that the induced subgraph G[S] on S is connected and r-regular. For r = 2, it is known that 2-MaxRICS is NP-hard and cannot be approximated within a factor of n1−ε in polynomial time for any ε > 0 if P 6 = NP . In this paper, we show ...

Given a graph G=(V,E) on n vertices, the MAXIMUM r-REGULAR INDUCED SUBGRAPH (M-r-RIS) problems ask for a maximum sized subset of vertices R ⊆ V such that the induced subgraph on R, G[R], is r-regular. We give an O(c) time algorithm for these problems for any fixed constant r, where c is a positive constant strictly less than 2, solving a well known open problem. These algorithms are then genera...

A graph is (t; r)-regular iit has at least one independent t-set of vertices and the open neighborhood of any such set contains exactly r vertices. Our goal is to show that when t¿ 3 and the order is su1ciently large, then the structure of (t; r)-regular graphs is similar to, but not exactly the same as the structure of (2; r)-regular graphs as derived by Faudree and Knisley. That is, there is ...

We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely generated Grothendieck categories. Applications are given for categories of comodules over a coalgebra and for categories of graded modules, and a link to the th...

Let R be an excellent local domain of positive characteristic with residue field k and let R+ be its absolute integral closure. If Tor1 (R +, k) vanishes, then R is weakly F-regular. If R has at most an isolated singularity or has dimension at most two, then R is regular.

let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

let r be an associative ring with unity. an element a in r is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von neumann) element in r. if every element of r is r-clean, then r is called an r-clean ring. in this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. further we prove that if 0 and 1 are the only idempotents...

in this paper we define a new type of rings ”almost powerhermitian rings” (a generalization of almost hermitian rings) and establish several sufficient conditions over a ring r such that, every regular matrix admits a diagonal power-reduction.