Large Girth Research Articles

Layered Subgraphs of the Hypercube

A subgraph of the $n$-dimensional hypercube is called 'layered' if it is a subgraph of a layer of some hypercube. In this paper we show that there exist subgraphs of the cube of arbitrarily large girth that are not layered. This answers a question of Axenovich, Martin and Winter. Perhaps surprisingly, these subgraphs may even be taken to be induced.

Graphs with Large Girth and Chromatic Number are Hard for Nullstellensatz

Graphs with Large Girth and Chromatic Number are Hard for Nullstellensatz

On the difference of the enhanced power graph and the power graph of a finite group

The difference graph D(G) of a finite group G is the difference of the enhanced power graph of G and the power graph of G, where all isolated vertices are removed. In this paper we study the connectedness and perfectness of D(G) with respect to various properties of the underlying group G. We also find several connections between the difference graph of G and the Gruenberg-Kegel graph of G. We also examine the operation of twin reduction on graphs, a technique which produces smaller graphs which may be easier to analyze. Applying this technique to simple groups can have a number of outcomes, not fully understood, but including some graphs with large girth.

On a class of column‐weight 3 decomposable LDPC codes with the analysis of elementary trapping sets

AbstractA column‐weight LDPC code with the parity‐check matrix is called decomposable if there exists a permutation on the rows of , such that can be decomposed into column‐weight one matrix. In this paper, some variations of edge coloring of graphs are used to construct some column‐weight three decomposable LDPC codes with girths at least six and eight. Applying the presented method on several known classes of bipartite graphs, some classes of column‐weight three decomposable LDPC codes are derived having flexibility in length and rate. Interestingly, the constructed parity‐check matrices based on the proper edge coloring of graphs can be considered as the base matrix of some high rate column‐weight three quasi‐cyclic (QC) LDPC codes with maximum‐achievable girth 20. The paper also leads to a simple characterization of elementary trapping sets of the decomposable codes based on the chromatic index of the corresponding normal graphs. This characterization corresponds to a simple search algorithm finds all possible existing elementary trapping sets in a girth‐6 or girth‐8 column‐weight 3 LDPC code which are layered super set of a short cycle in the Tanner graph of the code. Simulation results indicate that the QC‐LDPC codes with large girths lifted from the constructed base matrices have good performances over AWGN channel.

Inapproximability of Unique Games in Fixed-Point Logic with Counting

We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational numbers giving the approximate fraction of constraints that can be satisfied. We prove two new FPC-inexpressibility results for Unique Games: the existence of a $(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within any constant factor. Previous recent work has established similar FPC-inapproximability results for a small handful of other problems. Our construction builds upon some of these ideas, but contains a novel technique. While most FPC-inexpressibility results are based on variants of the CFI-construction, ours is significantly different. We start with a graph of very large girth and label the edges with random affine vector spaces over $\mathbb{F}_2$ that determine the constraints in the two structures. Duplicator's strategy involves maintaining a partial isomorphism over a minimal tree that spans the pebbled vertices of the graph.

On the girth of forbidden subgraphs of coloring graphs

The k-coloring graph of G, denoted Ck(G), is the graph whose vertex set is the proper k-colorings of the vertices of G, and where two k-colorings are adjacent if and only if they differ at exactly one vertex. A graph H is called permissible if there exists a graph G and a natural number k such that H is an induced subgraph of Ck(G), otherwise H is called forbidden. We show that every graph is either permissible or some subdivision of it is permissible. Furthermore, we prove if every edge in a triangle-free permissible graph is subdivided, the resulting graph is permissible, and all further subdivisions are also permissible. Although subdivisions tend to increase the girth of a graph, we conjecture that there exist forbidden subgraphs of arbitrarily large girth. Finally, we define forbidden subgraphs of girth seven and nine, the largest girths found so far.

The decycling number of a graph with large girth embedded in a surface

The decycling number of a graph with large girth embedded in a surface

On Colouring Oriented Graphs of Large Girth

We prove that for every oriented graph $D$ and every choice of positive integers $k$ and $\ell$, there exists an oriented graph $D^*$ along with a surjective homomorphism $\psi\colon D^* \to D$ such that: (i) girth$(D^*) \geq\ell$; (ii) for every oriented graph $C$ with at most $k$ vertices, there exists a homomorphism from $D^*$ to $C$ if and only if there exists a homomorphism from $D$ to C; and (iii) for every $D$-pointed oriented graph $C$ with at most $k$ vertices and for every homomorphism $\varphi\colon D^* \to C$ there exists a unique homomorphism $f\colon D \to C$ such that $\varphi=f \circ \psi$. Determining the chromatic number of an oriented graph $D$ is equivalent to finding the smallest integer $k$ such that $D$ admits a homomorphism to an order-$k$ tournament, so our main theorem yields results on the girth and chromatic number of oriented graphs. While our main proof is probabilistic (hence nonconstructive), for any given $\ell\geq 3$ and $k\geq 5$, we include a construction of an oriented graph with girth $\ell$ and chromatic number $k$.

Large girth and small oriented diameter graphs

In 2015, Dankelmann and Bau proved that for every connected bridgeless graph G of order n and minimum degree δ there is an orientation of diameter at most 11nδ+1+9. In 2016, Surmacs reduced this bound to 7nδ+1. In this paper, we consider the girth of a graph g and show that for any ε>0 there is a bound of the form (2g+ε)nh(δ,g)+O(g4ε2), where h(δ,g) is a polynomial. Fixing 0<ε<1, g, and δ gives an improvement on the result by Surmacs for sufficiently large n.

On an f-coloring generalization of linear arboricity of multigraphs

Given a multigraph G and a function f:V(G)→Z≥2 on its vertices, a degree-f subgraph of G is a spanning subgraph in which every vertex v has degree at most f(v). The degree-f arboricity af(G) of G is the minimum number of colors required to edge-color G into degree-f forests. For every constant function f and for every multigraph G, Truszczyński conjectured that af(G)≤max⁡{Δf(G)+1,a(G)}, where Δf(G)=maxv∈V(G)⁡⌈d(v)/f(v)⌉ and a(G) is the usual arboricity of G. This is a strong generalization of the Linear Arboricity Conjecture due to Akiyama, Exoo, and Harary. In this paper, we disprove Truszczyński's conjecture in a strong sense for general multigraphs. On the other hand, extending known results for linear arboricity, we prove that the conjecture holds for simple graphs with sufficiently large girth, and that it holds for all simple graphs asymptotically. More strongly, we prove these partial results in the setting of directed graphs, where the color classes are required to be analogously defined degree-f branchings.

Differential Effects of Clotrimazole on X-Ray Crystal Structures of Human Cytochromes P450 3A5 and 3A4.

Cytochromes P450 CYP3A5 and CYP3A4 exhibit differential plasticity that underlies differences in drug metabolism and drug-drug interactions. To extend previous studies, CYP3A4 and CYP3A5 were cocrystallized with clotrimazole, a compact ligand that binds to the heme iron in the catalytic center of the active site. Binding studies indicate that clotrimazole exhibits tight binding to CYP3A5 with a binding affinity (Kd) of <0.01 μM like that of CYP3A4. A single clotrimazole is bound to the heme iron in CYP3A4 that triggers expansion of active site cavity that reflects a loss of aromatic interactions between phenylalanine sidechains in the distal active site and increased conformational entropy for the F-F' connector due to reorientation of Phe-304 to accommodate clotrimazole. In contrast to CYP3A4, the CYP3A5 Phe-304 exhibits an induced fit along with Phe-213 to form edge-to-face aromatic interactions with heme-bound clotrimazole. These aromatic interactions between aromatic amino acids propagate by induced fits with a second clotrimazole residing in the distal active site and a third clotrimazole bound in an expanded entrance channel as well as between the three clotrimazoles. The large, expanded entrance channel surrounded by the C-terminal loop and the F' and A' helices in CYP3A5 suggests conformational selection for the binding of clotrimazole due to its large girth, which may also cause the entrance channel to remain open after the binding of the first clotrimazole to the heme iron. The additional binding sites suggest a path for sequential binding of one molecule to reach and bind to the heme iron. SIGNIFICANCE STATEMENT: Clotrimazole binds to the heme iron of CYP3A5 and CYP3A4. In CYP3A5, two clotrimazoles also bind in the distal active site and in an expanded entrance channel. Aromatic interactions between clotrimazoles and phenylalanine sidechains including Phe-304 indicate induced fits for each clotrimazole. In contrast to CYP3A5, displacement of the CYP3A4 Phe-304 rotamer by clotrimazole leads to extensive disruption of phenylalanine interactions that limit the space above the heme, to an expanded active site cavity, and to increased CYP3A4 conformational heterogeneity.

Disjointness graphs of short polygonal chains

The disjointness graph of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph G of any system of segments in the plane is χ-bounded, that is, its chromatic number χ(G) is upper bounded by a function of its clique number ω(G).Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have χ(G)≤(ω(G))3+ω(G).

Half-arc-transitive graphs of arbitrarily large girth

Half-arc-transitive graphs of arbitrarily large girth

A Study of the Activity of Adamantyl Amines against Mutant Influenza A M2 Channels Identified a Polycyclic Cage Amine Triple Blocker, Explored by Molecular Dynamics Simulations and Solid-State NMR.

We compared the anti-influenza potencies of 57 adamantyl amines and analogs against influenza A virus with serine-31 M2 proton channel, usually termed as WT M2 channel, which is amantadine sensitive. We also tested a subset of these compounds against viruses with the amantadine-resistant L26F, V27A, A30T, G34E M2 mutant channels. Four compounds inhibited WT M2 virus in vitro with mid-nanomolar potency, with 27 compounds showing sub-micromolar to low micromolar potency. Several compounds inhibited L26F M2 virus in vitro with sub-micromolar to low micromolar potency, but only three compounds blocked L26F M2-mediated proton current as determined by electrophysiology (EP). One compound was found to be a triple blocker of WT, L26F, V27A M2 channels by EP assays, but did not inhibit V27A M2 virus in vitro, and one compound inhibited WT, L26F, V27A M2 in vitro without blocking V27A M2 channel. One compound blocked only L26F M2 channel by EP, but did not inhibit virus replication. The triple blocker compound is as long as rimantadine, but could bind and block V27A M2 channel due to its larger girth as revealed by molecular dynamics simulations, while MAS NMR informed on the interaction of the compound with M2(18-60) WT or L26F or V27A.

Girth-based administered activity for pediatric 99m Tc-DMSA SPECT.

Pediatric molecular imaging requires a balance between administering an activity that will yield sufficient diagnostic image quality while maintaining patient radiation exposure at acceptable levels. In current clinical practice, this balance is arrived at by the current North American Consensus Guidelines in which patient weight is used to recommend the administered activity (AA). We have previously demonstrated that girth (waist circumference at the level of the kidneys) is better at equalizing image quality than patient weight for pediatric Tc-99m DMSA renal function imaging. However, the correlation between image quality (IQ), AA, and patient girth has not been rigorously and systematically developed. In this work, we generate a series of curves showing the tradeoff between AA and IQ as a function of patient girth, providing the data for standards bodies to develop the next generation of dosing guideline for pediatric DMSA SPECT. An anthropomorphic phantom series that included variations in age (5, 10, and 15 years), gender (M, F), local body morphometry (5, 10, 50, 90, and 95th girth percentiles), and kidney size (±15% standard size), was used to generate realistic SPECT projections. A fixed and clinically challenging defect-to-organ volume percentage (0.49% of renal cortex value) was used to model a focal defect with zero uptake (i.e., full local loss of renal function). Task-based IQ assessment methods were used to rigorously measure IQ in terms of renal perfusion defect detectability. This assessment was performed at multiple count levels (corresponding to various AAs) for groups of patients that had similar girths and defect sizes. Receiver-operating characteristics (ROC) analysis was applied; the area under the ROC curve (AUC) was used as a figure-of-merit for task performance. Curves showing the tradeoff between AUC and AA were generated for these groups of phantoms. Overall, the girth-based dosing method suggested different amounts of AA compared to weight-based dosing for the phantoms that had a relatively large body weight but a small girth or phantoms with relatively small bodyweight but large girth. Reductions of AA to 62.9% compared to weight-based dosing guidelines can potentially be realized while maintaining a baseline (AUC=0.80) IQ for certain 15-year-olds who have a relatively small girth and large defect size. Note that the task-based IQ results are heavily dependent on the simulated defect size for the defect detection task and the appropriate AUC value must be decided by the physicians for this diagnostic task. These results are based purely on simulation and are subject to future clinical validation. The study provides simulation-based IQ-AA data for a girth-based dosing method for pediatric renal SPECT, suggesting that patient waist circumference at the level of kidneys should be considered in selecting the AA needed to achieve an acceptable IQ. This data may be useful for standards bodies to develop girth-based dosing guidelines.

Giant component for the supercritical level‐set percolation of the Gaussian free field on regular expander graphs

AbstractWe consider the zero‐average Gaussian free field on a certain class of finite d‐regular graphs for fixed . This class includes d‐regular expanders of large girth and typical realisations of random d‐regular graphs. We show that the level set of the zero‐average Gaussian free field above level h has a giant component in the whole supercritical phase, that is for all , with probability tending to one as the size of the graphs tends to infinity. In addition, we show that this component is unique. This significantly improves the result of [4], where it was shown that a linear fraction of vertices is in mesoscopic components if , and together with the description of the subcritical phase from [4] establishes a fully‐fledged percolation phase transition for the model.

On the parameterized complexity of the acyclic matching problem

A matching is a set of edges in a graph with no common endpoint. A matching M is called acyclic if the induced subgraph on the endpoints of the edges in M is acyclic. Given a graph G and an integer k, Acyclic Matching problem seeks for an acyclic matching of size k in G. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs of maximum degree three and arbitrarily large girth. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter k. On the other hand, the problem is fixed parameter tractable with respect to the parameters treewidth, modular-width and (k,c4), where c4 is the number of cycles with length 4 of the input graph. We also prove that the problem is fixed parameter tractable with respect to the parameter k for the line graphs and every proper minor-closed class of graphs (including planar graphs).

Assessment of the variability of the morphological traits and differentiation of Cucurbita moschata in Cote d’Ivoire

With its predisposition to adapt to different environments, Cucurbita moschata grows well in various ecosystems. It is not a very exigent plant and has an inherent capacity for plasticity that underlies its large variability. An assessment of the accessions of C. moschata in Cote d’Ivoire shows that the plant exhibits large morphological and phenological variability for all the 28 measured traits. There are outliers among most measured traits. Further analysis indicates the emergence of three ecotypes in congruence with the three distinct ecosystems and their respective bioclimatic characteristics. In the savannah region, characterized by a short rainy season followed by a long dry season, a total yearly rainfall of 900 mm, an elevated daily temperature of 29 °C, and a high relative humidity of 80%, the cline of C. moschata is long and thin, with small leaves, small peduncles and small fruits. It has a high growth rate and accelerated phenology. The mountain region has a very long rainy season followed by a short dry season, a total pluviometry of 1400 mm, an average daily temperature of 27 °C and a relative humidity of 69%. The cline of C. moschata in the mountain region is characterized by a delayed flowering and a delayed fruit maturity, large number of small seeds and large fruits. The forest region has a favorable climate for the growth of C. moschata in Cote d’Ivoire. It has two rainy seasons that alternate with two dry seasons of unequal durations, an annual rainfall of 1200 mm, an average daily temperature of 27 °C and a relative humidity of 70%. The cline of C. moschata in that region has a large girth size, large dimensions of the leaves, long peduncles and larger and heavier fruits. The seeds are also large but in small number. It appears that the anatomy and physiology of the clines are differentiated primarily in response to soil water content and availability for the ontogeny of the plant.

Resolution of the Erdős–Sauer problem on regular subgraphs

Abstract In this paper, we completely resolve the well-known problem of Erdős and Sauer from 1975 which asks for the maximum number of edges an n-vertex graph can have without containing a k-regular subgraph, for some fixed integer $k\geq 3$ . We prove that any n-vertex graph with average degree at least $C_k\log \log n$ contains a k-regular subgraph. This matches the lower bound of Pyber, Rödl and Szemerédi and substantially improves an old result of Pyber, who showed that average degree at least $C_k\log n$ is enough. Our method can also be used to settle asymptotically a problem raised by Erdős and Simonovits in 1970 on almost regular subgraphs of sparse graphs and to make progress on the well-known question of Thomassen from 1983 on finding subgraphs with large girth and large average degree.

Subgraphs of Kneser graphs with large girth and large chromatic number

It is well known that for any integers k and g, there is a graph with chromatic number at least k and girth at least g. In 1960s, Erdős and Hajnal conjectured that for any k and g, there exists a number h(k,g), such that every graph with chromatic number at least h(k,g) contains a subgraph with chromatic number at least k and girth at least g. The Erdős-Hajnal Conjecture has been recently proposed in the setting of the fractional chromatic number. In support of the original and the modified conjecture, we prove that every Kneser graph, whose chromatic number is sufficiently large, contains a subgraph of large girth, whose (fractional) chromatic number is large.