Motivated by the work of Nešetřil and Rödl on “Partitions of vertices”, we are interested in obtaining some quantitative extensions of their result. In particular, given a natural number r and a graph G of order m with odd girth g, we show the existence of a graph H with odd girth at least g and order that is polynomial in m such that every r-coloring of the vertices of H yields a monochromatic...

Here we present an overview of various techniques performed concomitantly during penile prosthesis surgery to enhance penile length and girth. We report on the technique of ventral phalloplasty and its outcomes along with augmentation corporoplasty, suprapubic lipectomy, suspensory ligament release, and girth enhancement procedures. For the serious implanter, outcomes can be improved by combini...

let r be a fnite commutative ring and n(r) be the set of non unit elements of r. the non unit graph of r, denoted by gamma(r), is the graph obtained by setting all the elements of n(r) to be the vertices and defning distinct vertices x and y to be adjacent if and only if x - yin n(r). in this paper, the basic properties of gamma(r) are investigated and some characterization results regarding co...

In this paper, we propose a novel method for constructing quasi-cyclic low-density parity check (QC-LDPC) codes based row division method which can guarantee a concentrated node degree distribution with large girth. The main advantage is that large girth QC-LDPC codes can be easily constructed with a variety of block lengths and rates. Simulation results show that the proposed codes perform sig...

Design of binary LDPC codes with very low error floors is still a significant problem. Progressive Edge-Growth (P.E.G) original is a well known algorithm for constructing bipartite graphs with good girth properties. In this paper, we propose a characterization of the trapping set TS(5,3) and a cost function in P.E.G that allows to build regular girth 8 LDPC codes free of TS(5,3). We illustrate ...

We show that a graph of girth greater than 6 log k + 3 and minimum degree at least 3 has a minor of minimum degree greater than k. This is best possible up to a factor of at most 9/4. As a corollary, every graph of girth at least 6 log r + 3 log log r + c and minimum degree at least 3 has a Kr minor.

We give here new upper bounds on the size of a smallest feedback vertex set in planar graphs with high girth. In particular, we prove that a planar graph with girth g and size m has a feedback vertex set of size at most 4m 3g , improving the trivial bound of 2m g . We also prove that every 2-connected graph with maximum degree 3 and order n has a feedback vertex set of size at most n+2 3 .

This article presents a method for constructing large girth column-weight 2 QC-LDPC codes. A distance graph is first constructed using an existing method. The distance graph is then converted into a Tanner graph. The proposed method could easily construct codes with girths large than 12 and is more flexible compared to previous methods. Obtained codes show good bit error rate performance compar...

We prove that for all ` ≥ 3 and β > 0 there exists a sparse oriented graph of arbitrarily large order with oriented girth ` and such that any 1/2+β proportion of its arcs induces an oriented cycle of length `. As a corollary we get that there exist infinitely many oriented graphs with vanishing density of oriented girth ` such that deleting any 1/`-fraction of their edges does not destroy all t...

The girth of a graph G is the length of a shortest cycle in G. Dobson (1994, Ph.D. dissertation, Louisiana State University, Baton Rouge, LA) conjectured that every graph G with girth at least 2t+1 and minimum degree at least k t contains every tree T with k edges whose maximum degree does not exceed the minimum degree of G. The conjecture has been proved for t 3. In this paper, we prove Dobson...