Induced Copy Research Articles

SATURATION FOR THE BUTTERFLY POSET

Given a finite poset $\mathcal P$, we call a family $\mathcal F$ of subsets of $[n]$ $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The induced saturated number of $\mathcal P$, denoted by $\text{sat}^*(n,\mathcal P)$, is the size of the smallest $\mathcal P$-saturated family with ground set $[n]$. In this paper we are mainly interested in the four-point poset called the butterfly. Ferrara, Kay, Kramer, Martin, Reiniger, Smith and Sullivan showed that the saturation number for the butterfly lies between $\log_2{n}$ and $n^2$. We give a linear lower bound of $n+1$. We also prove some other results about the butterfly and the poset $\mathcal N$.

Two Erdős–Hajnal-type theorems in hypergraphs

The Erdős–Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph H, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of r-uniform hypergraphs.A theorem of Rödl asserts that if an n-vertex graph is non-universal then it contains an almost homogeneous set (i.e. one with edge density either very close to 0 or 1) of size Ω(n). We prove that if a 3-uniform hypergraph is non-universal then it contains an almost homogeneous set of size Ω(log⁡n). An example of Rödl from 1986 shows that this bound is tight.Let Rr(t) denote the size of the largest non-universal r-graph G so that neither G nor its complement contain a complete r-partite subgraph with parts of size t. We prove an Erdős–Hajnal-type stepping-up lemma, showing how to transform a lower bound for Rr(t) into a lower bound for Rr+1(t). As an application of this lemma, we improve a bound of Conlon–Fox–Sudakov by showing that R3(t)≥tΩ(t).

Constructions of infinite graphs with Ramsey property

AbstractFor every infinite cardinal λ and 2 ≤ n < ω there is a directed graph D of size λ such that D does not contain directed circuits of length ≤n and if its vertices are colored with <λ colors, then there is a monochromatic directed circuit of length n + 1. For every infinite cardinal λ and finite graph X there is a λ‐sized graph Y such that if the vertices of Y are colored with <λ colors, then there is a monochromatic induced copy of X. Further, Y does not contain larger cliques or shorter odd circuits than X. The constructions are using variants of Specker‐type graphs.

Induced saturation of [formula omitted

A graph G is called H-induced-saturated if G does not contain an induced copy of H, but removing any edge from G creates an induced copy of H and adding any edge of Gc to G creates an induced copy of H. Martin and Smith studied a related problem, and proved that there does not exist a P4-induced-saturated graph, where P4 is the path on 4 vertices. Axenovich and Csikós gave examples of families of graphs H for which H-induced-saturated graph G exists, and asked if there exists a Pn-induced-saturated graph when n≥5. Our aim in this short note is to show that there exists a P6-induced-saturated graph.

Design of a microaerobically inducible replicon for high-yield plasmid DNA production.

A pUC-derived replicon inducible by oxygen limitation was designed and tested in fed-batch cultures of Escherichia coli. It included the addition of a second inducible copy of rnaII, the positive replication control element. The rnaII gene was expressed from Ptrc and cloned into pUC18 to test the hypothesis that the ratio of the positive control molecule RNAII to the negative control element, RNAI, was the determinant of plasmid copy number per chromosome (PCN). The construct was evaluated in several E. coli strains. Evaluations of the RNAII/RNAI ratio, PCN and plasmid yield normalized to biomass (YpDNA/X ) were performed and the initial hypothesis was probed. Furthermore, in high cell-density cultures in shake flasks, an outstanding amount of 126 mg/L of plasmid was produced. The microaerobically inducible plasmid was obtained by cloning the rnaII gene under the control of the oxygen-responsive Vitreoscilla stercoraria hemoglobin promoter. For this plasmid, but not for pUC18, the RNAII/RNAI ratio, PCN and YpDNA/X efficiently increased after the shift to the microaerobic regime in fed-batch cultures in a 1 L bioreactor. The YpDNA/X of the inducible plasmid reached 12 mg/g at the end of the fed-batch but the original pUC18 only reached ca. 6 mg/g. The proposed plasmid is a valuable alternative for the operation and scale-up of plasmid DNA production processes in which mass transfer limitations will not represent an issue.

On Betti numbers of flag complexes with forbidden induced subgraphs

AbstractWe analyse the asymptotic extremal growth rate of the Betti numbers of clique complexes of graphs on n vertices not containing a fixed forbidden induced subgraph H.In particular, we prove a theorem of the alternative: for any H the growth rate achieves exactly one of five possible exponentials, that is, independent of the field of coefficients, the nth root of the maximal total Betti number over n-vertex graphs with no induced copy of H has a limit, as n tends to infinity, and, ranging over all H, exactly five different limits are attained.For the interesting case where H is the 4-cycle, the above limit is 1, and we prove a superpolynomial upper bound.

The non-flavonoid inducible nodA3 and the flavonoid regulated nodA1 genes of Rhizobium tropici CIAT 899 guarantee nod factor production and nodulation of different host legumes

The symbiosis between rhizobia and their host legumes is initiated by a complex molecular dialogue in which the activation of bacterial NodD proteins by appropriate plant flavonoids triggers the expression of the bacterial nodulation (nod) genes. These genes are involved in the synthesis and export of the Nodulation factors (NF), which are signal molecules that, when recognized by plant receptors, launch the symbiotic process. The core of NF is synthesized by proteins encoded by the nodABC genes, whereas the rest of the nod genes protein products are involved in the decoration of NF with different chemical substituents and their export to the environment. Rhizobium tropici CIAT 899, the microsymbiont of common bean (Phaseolus vulgaris), is characterized for tolerating multiple physical stresses and for synthesizing a large variety of NF not only in the presence of inducing flavonoids but also when high concentrations of salt are present. In addition, another interesting feature of the R. tropici CIAT 899 genome is the presence of three different nodA genes on the symbiotic plasmid, although their exact roles remain to be elucidated. In this work, we characterize the role of the three NodA proteins of R. tropici CIAT 899 in symbiosis. We have analyzed by quantitative PCR the expression of the nodA1, nodA2 and nodA3 genes of CIAT 899, showing that only nodA1 and nodA2 are inducible by flavonoids. We have also constructed single, double and triple non-polar mutants in these genes in order to study their importance in NF production and in nodulation of four different host legumes of CIAT 899: P. vulgaris, Leucaena leucocephala, Lotus japonicus, and L. burttii. The nodA1 and nodA2 genes of CIAT 899 (both inducible by flavonoids) are more related between them than with nodA3 (non-flavonoid inducible). Interestingly, the presence of NodA1 or NodA3 is crucial for nodulation with L. leucocephala and L. japonicus, since both NodA1 or NodA3 guarantee Nod factor production upon apigenin induction. Interestingly, osmotic-stressing conditions increase the quantity and diversity of the Nod factors synthesized by NodA2, being these molecules able to induce the formation of nodule primordia on P. vulgaris. We concluded that R. tropici CIAT 899 requires at least the flavonoid-induced nodA1 or the flavonoid-independent nodA3 genes for ensuring symbiotic success in the four assayed host-legumes. The inducible nodA2 gene is sufficient to induce nodulation on P. vulgaris and L. burttii but not in L. leucocephala and L. japonicus. To our knowledge, in this work we report for the first time that a non-flavonoid inducible copy of nodA is enough to induce nodulation on legumes.

Degenerate Turán Problems for Hereditary Properties

Let $H$ be a graph and $t\geqslant s\geqslant 2$ be integers. We prove that if $G$ is an $n$-vertex graph with no copy of $H$ and no induced copy of $K_{s,t}$, then $\lambda(G) = O\left(n^{1-1/s}\right)$ where $\lambda(G)$ is the spectral radius of the adjacency matrix of $G$. Our results are motivated by results of Babai, Guiduli, and Nikiforov bounding the maximum spectral radius of a graph with no copy (not necessarily induced) of $K_{s,t}$.

Forbidden induced subposets of given height

Let P be a partially ordered set. The function La⁎(n,P) denotes the size of the largest family F⊂2[n] that does not contain an induced copy of P. It was proved by Methuku and Pálvölgyi that there exists a constant CP (depending only on P) such that La⁎(n,P)<CP(n⌊n/2⌋). However, the order of the constant CP following from their proof is typically exponential in |P|. Here, we show that if the height of the poset is constant, this can be improved. We show that for every positive integer h there exists a constant ch such that if P has height at most h, thenLa⁎(n,P)≤|P|ch(n⌊n/2⌋).Our methods also immediately imply that similar bounds hold in grids as well. That is, we show that if F⊂[k]n such that F does not contain an induced copy of P and n≥2|P|, then|F|≤|P|chw, where w is the width of [k]n.A small part of our proof is to partition 2[n] (or [k]n) into certain fixed dimensional grids of large sides. We show that this special partition can be used to derive bounds in a number of other extremal set theoretical problems and their generalizations in grids, such as the size of families avoiding weak posets, Boolean algebras, or two distinct sets and their union. This might be of independent interest.

Turán Number of an Induced Complete Bipartite Graph Plus an Odd Cycle

Let k ⩾ 2 be an integer. We show that if s = 2 and t ⩾ 2, or s = t = 3, then the maximum possible number of edges in a C2k+1-free graph containing no induced copy of Ks,t is asymptotically equal to (t − s + 1)1/s(n/2)2−1/s except when k = s = t = 2.This strengthens a result of Allen, Keevash, Sudakov and Verstraëte [1], and answers a question of Loh, Tait, Timmons and Zhou [14].

On Perfect Matchings and Tilings in Uniform Hypergraphs

In this paper we study some variants of Dirac-type problems in hypergraphs. First, we show that for $k\ge 3$, if $H$ is a $k$-graph on $n\in k\mathbb N$ vertices with independence number at most $n/p$ and minimum codegree at least $(1/p+o(1))n$, where $p$ is the smallest prime factor of $k$, then $H$ contains a perfect matching. Second, we show that if $H$ is a $3$-graph on $n\in 3\mathbb N$ vertices which does not contain any induced copy of $K_4^-$ (the unique $3$-graph with $4$ vertices and $3$ edges) and has minimum codegree at least $(1/3+o(1)))n$, then $H$ contains a perfect matching. Moreover, if we allow the matching to miss at most $3$ vertices, then the minimum degree condition can be reduced to $(1/6+o(1)))n$. Third, we show that if $H$ is a $3$-graph on $n\in 4\mathbb N$ vertices which does not contain any induced copy of $K_4^-$ and has minimum codegree at least $(1/8+o(1)))n$, then $H$ contains a perfect $Y$-tiling, where $Y$ represents the unique $3$-graph with $4$ vertices and $2$ edges. We also provide the examples showing that our minimum codegree conditions are asymptotically best possible. Our main tool for finding the perfect matching is a characterization theorem that characterizes the $k$-graphs with minimum codegree at least $n/k$ which contain a perfect matching.

A LYM inequality for induced posets

We prove that for every partially ordered set P, there exists a constant c(P) such that every family F of subsets of [n] ordered by inclusion which contains no induced copy of P satisfies ∑F∈F1/(n|F|)≤c(P).

Induced Turán Numbers

The classical Kővári–Sós–Turán theorem states that ifGis ann-vertex graph with no copy ofKs,tas a subgraph, then the number of edges inGis at mostO(n2−1/s). We prove that if one forbidsKs,tas aninducedsubgraph, and also forbidsanyfixed graphHas a (not necessarily induced) subgraph, the same asymptotic upper bound still holds, with different constant factors. This introduces a non-trivial angle from which to generalize Turán theory to induced forbidden subgraphs, which this paper explores. Along the way, we derive a non-trivial upper bound on the number of cliques of fixed order in aKr-free graph with no induced copy ofKs,t. This result is an induced analogue of a recent theorem of Alon and Shikhelman and is of independent interest.

Generation of Conditional Mutants in Borrelia burgdorferi.

Mutational studies aimed at characterizing the function(s) of bacterial genes required for growth or viability are constrained by the inability to generate deletion strains lacking the gene of interest. To circumvent this limitation, it is possible to generate conditional mutants in which a copy of the gene of interest is introduced into the bacteria to compensate for the loss of the native allele. Expression of the non-native copy of the target gene is typically under control of an inducible promoter, which allows for controllable and regulated production of the gene of interest. Conditional mutants are also broadly useful for phenotypic analyses of genes that require a tightly regulated and artificially inducible copy of the target gene. Herein, we describe the methods used to generate and confirm conditional mutant clones in Borrelia burgdorferi utilizing the Borrelia-adapted lac operator/repressor system.

The Structure of Typical Eye-Free Graphs and a Turán-Type Result for Two Weighted Colours

The (a,b)-eye is the graph Ia,b = Ka+b \ Kb obtained by deleting the edges of a clique of size b from a clique of size a+b. We show that for any a,b ≥ 2 and p ∈ (0,1), if we condition the random graph G ~ G(n,p) on having no induced copy of Ia,b, then with high probability G is close to an a-partite graph or the complement of a (b−1)-partite graph. Our proof uses the recently developed theory of hypergraph containers, and a stability result for an extremal problem with two weighted colours. We also apply the stability method to obtain an exact Turán-type result for this extremal problem.

Forbidden pairs of disconnected graphs for traceability in connected graphs

Let H be a class of given graphs. A graph G is said to be H-free if G contains no induced copies of H for every H?H. A graph is called traceable if it contains a Hamilton path. Faudree and Gould (1997) characterized all the pairs {R,S} of connected subgraphs such that every connected {R,S}-free graph is traceable. Li and Vrána (2016) first consider the disconnected forbidden subgraphs, and characterized all pairs of disconnected forbidden subgraphs for hamiltonicity. In this article, we characterize all pairs {R,S} of graphs such that there exists an integer n0 such that every connected {R,S}-free graph of order at least n0 is traceable.

Dichotomy Results on the Hardness of $H$-free Edge Modification Problems

For a graph $H$, the $H$-free Edge Deletion problem asks whether there exist at most $k$ edges whose deletion from the input graph $G$ results in a graph without any induced copy of $H$. $H$-free Edge Completion and $H$-free Edge Editing are defined similarly where only completion (addition) of edges are allowed in the former and both completion and deletion are allowed in the latter. We completely settle the classical complexities of these problems by proving that $H$-free Edge Deletion is NP-complete if and only if $H$ is a graph with at least two edges, $H$-free Edge Completion is NP-complete if and only if $H$ is a graph with at least two nonedges, and $H$-free Edge Editing is NP-complete if and only if $H$ is a graph with at least three vertices. Our result on $H$-free Edge Editing resolves a conjecture by Alon and Stav [Theoret. Comput. Sci., 2009, pp. 4920--4927]. Additionally, we prove that these NP-complete problems cannot be solved in parameterized subexponential time, i.e., in time $2^{o(k)}\cd...

Graphs with Induced-Saturation Number Zero

Given graphs $G$ and $H$, $G$ is $H$-saturated if $H$ is not a subgraph of $G$, but for all $e \notin E(G)$, $H$ appears as a subgraph of $G + e$. While for every $n \ge |V(H)|$, there exists an $n$-vertex graph that is $H$-saturated, the same does not hold for induced subgraphs. That is, there exist graphs $H$ and values of $n \ge |V(H)|$, for which every $n$-vertex graph $G$ either contains $H$ as an induced subgraph, or there exists $e \notin E(G)$ such that $G + e$ does not contain $H$ as an induced subgraph. To circumvent this Martin and Smith make use of a generalized notion of "graph" when introducing the concept of induced saturation and the induced saturation number of graphs. This allows for edges that can be included or excluded when searching for an induced copy of $H$, and the induced saturation number is the minimum number of such edges that are required.In this paper, we show that the induced saturation number of many common graphs is zero. This yields graphs that are $H$-induced-saturated. That is, graphs such that no induced copy of $H$ exists, but adding or deleting any edge creates an induced copy of $H$. We introduce a new parameter for such graphs, indsat*($n;H$), which is the minimum number of edges in an $H$-induced-saturated graph. We provide bounds on indsat*($n;H$) for many graphs. In particular, we determine indsat*($n;H$) completely when $H$ is the paw graph $K_{1,3}+e$, and we determine indsat*(n;$K_{1,3}$) within an additive constant of four.

Forbidden Pairs of Disconnected Graphs Implying Hamiltonicity

Let H be a given graph. A graph G is said to be H-free if G contains no induced copies of H. For a class of graphs, the graph G is -free if G is H-free for every . Bedrossian characterized all the pairs of connected subgraphs such that every 2-connected -free graph is hamiltonian. Faudree and Gould extended Bedrossian's result by proving the necessity part of the result based on infinite families of non-hamiltonian graphs. In this article, we characterize all pairs of (not necessarily connected) graphs such that there exists an integer n0 such that every 2-connected -free graph of order at least n0 is hamiltonian.

The Erdös--Hajnal Conjecture for Long Holes and Antiholes

Erdos and Hajnal conjectured that for every graph $H$, there exists a constant $c_H$ such that every graph $G$ on $n$ vertices which does not contain an induced copy of $H$ has a clique or a stable set of size $n^{c_H}$. We prove that for every $k$ there exists $c_k>0$ such that every graph $G$ on $n$ vertices not inducing a cycle of length at least $k$ nor its complement contains a clique or a stable set of size at least $n^{c_k}$.