Large Girth Research Articles

Incidence Posets and Cover Graphs

We prove two theorems concerning incidence posets of graphs, cover graphs of posets and a related graph parameter. First, answering a question of Haxell, we show that the chromatic number of a graph is not bounded in terms of the dimension of its incidence poset, provided the dimension is at least four. Second, answering a question of Kříž and Nešetřil, we show that there are graphs with large girth and large chromatic number among the class of graphs having eye parameter at most two.

Van der Waerden function and colorings of hypergraphs with large girth

Van der Waerden function and colorings of hypergraphs with large girth

On the Girth of Quasi-Cyclic Protograph LDPC Codes

In this paper, we study the relationships between the girth of the Tanner graph of a quasi-cyclic (QC) protograph low-density parity-check (LDPC) code, the lifting degree, and the size and the structure of the base graph. As a result, for a given base graph, we derive a lower bound on the lifting degree as a necessary condition for the lifted graph to have a certain girth. This also provides an upper bound on the girth of the family of graphs lifted from a given base graph with a given lifting degree. The upper bounds derived here, which are applicable to both regular and irregular base graphs with no parallel edges, are in some cases more general and in some other cases tighter than the existing bounds. The results presented in this work can be used to design cyclic liftings with relatively small degree and relatively large girth. As an example, we present new QC protograph LDPC code constructions with girth 8 using fully connected base graphs. These constructions provide upper bounds on the lifting degree required for achieving girth 8 using fully connected base graphs.

Patient's Body Size Influences Dental Hygienist Shoulder Kinematics

OCCUPATIONAL APPLICATIONS This study found that dental hygienists are exposed to higher shoulder elevation angles when treating a simulated big-girth patient versus a simulated average-girth patient. These differences in motion patterns may increase dental hygienists’ susceptibility to shoulder musculoskeletal disorders. Ergonomic interventions may be needed to facilitate solutions to problems associated with treating patients who are overweight or obese. Ergonomics and human factors practitioners should educate dental hygienists to be more aware of their body posture, specifically shoulder position, while working on patients with big chest girth. TECHNICAL ABSTRACT Background: Dental hygienists suffer from musculoskeletal disorders of the upper extremities, and the prevalence of these disorders increases with years of occupation. Furthermore, the prevalence of overweight and obese individuals in the United States has been rapidly increasing for both adults and children in recent years. This may increase the involvement of dental hygienists with obese patients, which may result in an increased prevalence of shoulder musculoskeletal disorders. The effect of patient's body size on the three-dimensional humeral and scapular kinematics of dental hygienists, however, is currently unknown. Purpose: The aim of the current study was to measure the influence of patient girth on the shoulder kinematics of dental hygienists during teeth instrumentation in a simulated work environment. Method: Sixteen female dental hygienists participated in a lab-based study that simulated a dental hygienist workplace. Three-dimensional kinematic data were collected for the right and left scapula and humerus using the Polhemus Liberty magnetic tracking system (Colchester, VT, USA). To quantify differences in humeral and scapular kinematics and exposure variables, two independent variables with two levels were chosen: handedness and body type size. Separate two-way ANOVAs with repeated measures were conducted. Results: Dental hygienists sustained significant higher mean humeral elevation angles and higher mean scapular upward rotation angles for the dominant and non-dominant hands while instrumenting a simulated patient with big girth chest. Furthermore, dental hygienists spent significantly more time above 30° and 60° of humeral elevation for the dominant and non-dominant hands while instrumenting this simulated patient. Conclusion: This study provided evidence that dental hygienists alter their shoulder kinematics while instrumenting patients having a larger girth. Consequently, working with patients who are obese, or with a larger body size, may increase dental hygienist susceptibility to shoulder musculoskeletal disorders.

Coloring Non-uniform Hypergraphs Without Short Cycles

The work deals with a generalization of Erd�s---Lovasz problem concerning colorings of non-uniform hypergraphs. Let H = (V, E) be a hypergraph and let $${{f_r(H)=\sum\limits_{e \in E}r^{1-|e|}}}$$ f r ( H ) = � e � E r 1 - | e | for some r � 2. Erd�s and Lovasz proposed to find the value f (n) equal to the minimum possible value of f 2(H) where H is 3-chromatic hypergraph with minimum edge-cardinality n. In the paper we study similar problem for the class of hypergraphs with large girth. We prove that if H is a hypergraph with minimum edge-cardinality n � 3 and girth at least 4, satisfying the inequality $$f_r(H) \leq \frac{1}{2}\, \left(\frac{n}{{\rm ln}\, n}\right)^{2/3},$$ f r ( H ) ≤ 1 2 n ln n 2 / 3 , then H is r -colorable. Our result improves previous lower bounds for f (n) in the class of hypergraphs without 2- and 3-cycles.

Backbone coloring for graphs with large girths

For a graph G and a subgraph H (called a backbone graph) of G, a backbonek-coloring ofGwith respect toH is a proper vertex coloring of G using colors from the set {1,2,…,k}, with an additional condition that colors for any two adjacent vertices in H must differ by at least two. The backbone chromatic number ofGoverH, denoted by BBC(G,H), is the smallest k of a backbone k-coloring admitted by G with respect to H. Broersma, Fomin, Golovach, and Woeginger (2007) [2] showed that BBC(G,H)≤2χ(G)−1 holds for every G and H; moreover, for every n there exists a graph G with a spanning tree T such that χ(G)=n and the bound is sharp. To answer a question raised in Broersma et al. (2007) [2], Miškuf, Škrekovski, and Tancer (2009) [16] proved that for any n there exists a triangle-free graph G with a spanning tree T such that χ(G)=n and BBC(G,T)=2n−1. We extend this result by showing that for any positive integers n and l, there exists a graph G with a spanning tree T such that G has girth at least l, χ(G)=n, and BBC(G,T)=2n−1.

Constraints, MMSNP and expander relational structures

We give a poly-time construction for a combinatorial classic known as Sparse Incomparability Lemma, studied by Erdős, Lovász, Nešetřil, Rödl and others: We show that every Constraint Satisfaction Problem is poly-time equivalent to its restriction to structures with large girth. This implies that the complexity classes CSP and Monotone Monadic Strict NP introduced by Feder and Vardi are computationally equivalent. The technical novelty of the paper is a concept of expander relations and a new type of product for relational structures: a generalization of the zig-zag product, the twisted product.

Uniform local amenability

The main results of this paper show that various coarse (‘large scale’) geometric properties are closely related. In particular, we show that property A implies the operator norm localisation property, and thus that norms of operators associated to a very large class of metric spaces can be effectively estimated. The main tool is a new property called uniform local amenability. This property is easy to negate, which we use to study some ‘bad’ spaces: specifically, expanders and graphs with large girth. We also generalise and reprove a theorem of Nowak relating amenability and asymptotic dimension in the quantitative setting.

Visual Diagnosis: Infant With Growth Failure, Body Asymmetry, and Dysmorphic Features

1. Deepa Prasad, MD* 2. Valerie Navarrete, MD† 3. Srividya Naganathan, MD, FAAP‡ 1. *Chief Resident, Department of Pediatrics, Jersey Shore University Medical Center, Neptune, NJ. 2. †Pediatric resident, Jersey Shore University Medical Center, Neptune, NJ. 3. ‡Assistant professor of Pediatrics, UMDNJ, Pediatric clerkship site director, Jersey Shore University Medical Center, Neptune, NJ. A 5-month-old girl is admitted to the hospital by her pediatrician for evaluation of failure to thrive. The mother reports that her daughter has always been small and that she feels her daughter is not gaining weight as expected despite feeding her 24 ounces of a 22-calorie-per-ounce premature formula each day (135 kcal/kg per day). The mother also expresses concern that her daughter has a large head, developmental delays, and a right leg that is growing longer than the left. Evaluation by an orthopedic surgeon found no evidence of hip dysplasia. The infant’s past medical history reveals an uncomplicated premature birth at 35 weeks’ gestation via vaginal delivery. Her birthweight was 2 pounds and 14 ounces (<2 standard deviations [SD] from mean), which plotted as small for gestational age. She was hospitalized for 3 weeks for feeding and growth monitoring. Review of systems is negative for vomiting, diarrhea, swallowing difficulties, or weight loss. She is not on any medications, and her immunizations are up to date. She is an only child and there is no history of consanguinity. Family history is unremarkable. Physical examination reveals an alert, active, thin baby whose vital signs are normal for age. Her weight is 3.9 kg (<-3 SD from mean, according to World Health Organization Growth charts), length is 21 inches (<-3 SD from mean), head circumference is 16 inches (between 0 and -1 SD). Her head appears large in relation to the rest of her body, and she has a prominent forehead ( Fig 1 ) . Her right leg has larger girth and is 1 inch longer than the left leg ( Fig 2 ) . Further examination of her extremities reveals clinodactyly of her fifth fingers ( Fig 3 ) . She has poor head control and mild generalized hypotonia. The remainder of her …

Asymptotic Lower Bounds on Circular Chromatic Index of Snarks

We prove that the circular chromatic index of a cubic graph $G$ with $2k$ vertices and chromatic index $4$ is at least $3+2/k$. This bound is (asymptotically) optimal for an infinite class of cubic graphs containing bridges. We also show that the constant $2$ in the above bound can be increased for graphs with larger girth or higher connectivity. In particular, if $G$ has girth at least $5$, its circular chromatic index is at least $3+2.5/k$. Our method gives an alternative proof that the circular chromatic index of the generalised type 1 Blanuša snark $B_m^1$ is $3+2/3m$.

Unicyclic graphs of minimal spectral radius

It was conjectured by Li and Feng in 1979 that the unicyclic graph formed by a cycle of order g linking to an endvertex of a path of length k minimizes the spectral radius of all unicyclic graphs of order g+k and girth g. In 1987, Cao proved that this conjecture is true for k ≥ g(g − 2)/8 and false for k = 2 and sufficiently large g. In this note, we show that g > 12 suffices for the counterexample and give more counterexamples with large girth for any integer k > 1.

Uniquely D-colourable Digraphs with Large Girth

Abstract Let C and D be digraphs. A mapping f : V(D) →V(C) is a C-colouring if for every arc uv of D, either f (u) f (v) is an arc of C or f (u) = f (v), and the preimage of every vertex of C induces an acyclic subdigraph in D. We say that D is C-colourable if it admits a C-colouring and that D is uniquely C-colourable if it is surjectively C-colourable and any two C-colourings of D differ by an automorphism of C. We prove that if a digraph D is not C-colourable, then there exist digraphs of arbitrarily large girth that are D-colourable but not C-colourable. Moreover, for every digraph D that is uniquely D-colourable, there exists a uniquely D-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number r ≥ 1, there are uniquely circularly r-colourable digraphs with arbitrarily large girth.

Design of Multiple-Edge Protographs for QC LDPC Codes Avoiding Short Inevitable Cycles

There have been lots of efforts on the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with large girth. However, most of them focus on protographs with single edges and little research has been done for the construction of QC LDPC codes lifted from protographs with multiple (i.e., parallel) edges. Compared to single-edge protographs, multiple-edge protographs have benefits such that QC LDPC codes lifted from them can potentially have larger minimum Hamming distance. In this paper, all subgraph patterns of multiple-edge protographs, which prevent QC LDPC codes from having large girth by inducing inevitable cycles, are fully investigated based on a graph-theoretic approach. By using combinatorial designs, a systematic construction method of multiple-edge protographs is proposed for regular QC LDPC codes with girth at least 12 and another method is proposed for regular QC LDPC codes with girth at least 14. Moreover, a construction algorithm of QC LDPC codes based on certain liftings of multiple-edge protographs is proposed and it is shown that the resulting QC LDPC codes have larger upper bounds on the minimum Hamming distance than those lifted from single-edge protographs. Simulation results are provided to compare the performance of the proposed QC LDPC codes with progressive edge-growth (PEG) LDPC codes and with PEG QC LDPC codes.

Maximum edge‐cuts in cubic graphs with large girth and in random cubic graphs

AbstractWe show that for every cubic graph Gwith sufficiently large girth there exists a probability distribution on edge‐cuts in Gsuch that each edge is in a randomly chosen cut with probability at least 0.88672. This implies that Gcontains an edge‐cut of size at least 1.33008n, where nis the number of vertices of G, and has fractional cut covering number at most 1.127752. The lower bound on the size of maximum edge‐cut also applies to random cubic graphs. Specifically, a random n‐vertex cubic graph a.a.s. contains an edge‐cut of size 1.33008n‐ o(n). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012

On the optimum signal constellation design for high-speed optical transport networks

In this paper, we first describe an optimum signal constellation design algorithm, which is optimum in MMSE-sense, called MMSE-OSCD, for channel capacity achieving source distribution. Secondly, we introduce a feedback channel capacity inspired optimum signal constellation design (FCC-OSCD) to further improve the performance of MMSE-OSCD, inspired by the fact that feedback channel capacity is higher than that of systems without feedback. The constellations obtained by FCC-OSCD are, however, OSNR dependent. The optimization is jointly performed together with regular quasi-cyclic low-density parity-check (LDPC) code design. Such obtained coded-modulation scheme, in combination with polarization-multiplexing, is suitable as both 400 Gb/s and multi-Tb/s optical transport enabling technology. Using large girth LDPC code, we demonstrate by Monte Carlo simulations that a 32-ary signal constellation, obtained by FCC-OSCD, outperforms previously proposed optimized 32-ary CIPQ signal constellation by 0.8 dB at BER of 10(-7). On the other hand, the LDPC-coded 16-ary FCC-OSCD outperforms 16-QAM by 1.15 dB at the same BER.

Identifying Codes in Line Graphs

AbstractAn identifying code of a graph is a subset of its vertices such that every vertex of the graph is uniquely identified by the set of its neighbors within the code. We study the edge‐identifying code problem, i.e. the identifying code problem in line graphs. If denotes the size of a minimum identifying code of an identifiable graph G, we show that the usual bound , where n denotes the order of G, can be improved to in the class of line graphs. Moreover, this bound is tight. We also prove that the upper bound , where is the line graph of G, holds (with two exceptions). This implies that a conjecture of R. Klasing, A. Kosowski, A. Raspaud, and the first author holds for a subclass of line graphs. Finally, we show that the edge‐identifying code problem is NP‐complete, even for the class of planar bipartite graphs of maximum degree 3 and arbitrarily large girth.

Acyclic edge coloring of graphs with large girths

A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G, denoted by x′a(G), is the least number of colors such that G has an acyclic edge k-coloring. Let G be a graph with maximum degree Δ and girth g(G), and let 1 ⩽ r ⩽ 2Δ be an integer. In this paper, it is shown that there exists a constant c > 0 such that if \(g(G) \geqslant \frac{{c\Delta }} {r}\log \left( {\Delta ^2 /r} \right)\) then x′a(G) ⩽ Δ+r + 1, which generalizes the result of Alon et al. in 2001. When G is restricted to series-parallel graphs, it is proved that x′a(G) = Δ if Δ ⩾ 4 and g(G) ⩾ 4; or Δ ⩾ 3 and g(G) ⩾ 5.

Locally Identifying Coloring of Graphs

We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring $c$ of a graph $G$ is said to be locally identifying, if for any adjacent vertices $u$ and $v$ with distinct closed neighborhoods, the sets of colors that appear in the closed neighborhood of $u$ and $v$, respectively, are distinct. Let $\chi_{\rm{lid}}(G)$ be the minimum number of colors used in a locally identifying vertex-coloring of $G$. In this paper, we give several bounds on $\chi_{\rm{lid}}$ for different families of graphs (planar graphs, some subclasses of perfect graphs, graphs with bounded maximum degree) and prove that deciding whether $\chi_{\rm{lid}}(G)=3$ for a subcubic bipartite graph $G$ with large girth is an NP-complete problem.

An improved upper bound for the bondage number of graphs on surfaces

The bondage number b(G) of a graph G is the smallest number of edges whose removal from G results in a graph with larger domination number. Recently Gagarin and Zverovich showed that, for a graph G with maximum degree Δ(G) and embeddable on an orientable surface of genus h and a non-orientable surface of genus k, b(G)≤min{Δ(G)+h+2,Δ+k+1}. They also gave examples showing that adjustments of their proofs implicitly provide better results for larger values of h and k. In this paper we establish an improved explicit upper bound for b(G), using the Euler characteristic χ instead of the genera h and k, with the relations χ=2−2h and χ=2−k. We show that b(G)≤Δ(G)+⌊r⌋ for the case χ≤0 (i.e. h≥1 or k≥2), where r is the largest real root of the cubic equation z3+2z2+(6χ−7)z+18χ−24=0. Our proof is based on the technique developed by Carlson–Develin and Gagarin–Zverovich, and includes some elementary calculus as a new ingredient. We also find an asymptotically equivalent result b(G)≤Δ(G)+⌈12−6χ−1/2⌉ for χ≤0, and a further improvement for graphs with large girth.

On Families of Graphs of Large Cycle Indicator, Matrices of Large Order and Key Exchange Protocols With Nonlinear Polynomial Maps of Small Degree

This paper studies the group theoretical protocol of Diffie–Hellman key exchange in the case of symmetrical group \({S_{p^n}}\) and more general Cremona group \({C(\mathbb K^n)}\) of polynomial automorphisms of free module \({\mathbb K^n}\) over arbitrary commutative ring \({\mathbb K}\). This algorithm depends very much on the choice of the base \({g_n \in C( \mathbb K^n)}\). It is important to work with the base \({g_n \in C( \mathbb K^n)}\), which is a polynomial map of a small degree and a large order such that the degrees of all powers \({g_n^k}\) are also bounded by a small constant. We suggest fast algorithms for generation of a map \({g_n={f_n} \xi_nf_n^{-1}}\), where ξn is an affine transformation (degree is 1) of a large order and fn is a fixed nonlinear polynomial map in n variables such that \({f_n^{-1}}\) is also a polynomial map and both maps fn and \({f_n^{-1}}\) are of small degrees. The method is based on properties of infinite families of graphs with a large cycle indicator and families of graphs of a large girth in particular. It guaranties that the order of gn is tending to infinity as the dimension n tends to infinity. We propose methods of fast generation of special families of cubical maps fn such that \({f_n^{-1}}\) is also of degree 3 based on properties of families of graphs of a large girth and graphs with a large cycle indicator. At the end we discuss cryptographical applications of maps of the kind τ fn τ−1 and some graph theoretical problems motivated by such applications.