Showing papers in "Combinatorics, Probability & Computing in 2012"
TL;DR: In this paper, a review article focusing on conducting polymers and their applications is presented, which comprises two main parts of investigation: the first part is focusing on conducting polymers (polythiophene, polyparaphenylene vinylene, polycarbazole, polyaniline, and polypyrrole) and the second part regards their applications, such as Supercapacitors, Light emitting diodes (LEDs), Solar cells, Field effect transistor (FET), and Biosensors).
Abstract: This review article focuses on conducting polymers and their applications. Conducting polymers (CPs) are an exciting new class of electronic materials, which have attracted an increasing interest since their discovery in 1977. They have many advantages, as compared to the non-conducting polymers, which is primarily due to their electronic and optic properties. Also, they have been used in artificial muscles, fabrication of electronic device, solar energy conversion, rechargeable batteries, and sensors. This study comprises two main parts of investigation. The first focuses conducting polymers (polythiophene, polyparaphenylene vinylene, polycarbazole, polyaniline, and polypyrrole). The second regards their applications, such as Supercapacitors, Light emitting diodes (LEDs), Solar cells, Field effect transistor (FET), and Biosensors. Both parts have been concluded and summarized with recent reviewed 233 references.
157 citations
TL;DR: This paper verifies the conjecture that for any $t < \frac{n}{3k^2}$, every k-uniform hypergraph on n vertices without t disjoint edges has at most max $binom{kt-1}{k}-\binom-n-t-t+1-k$ edges.
Abstract: More than forty years ago, ErdAs conjectured that for any $t \leq \frac{n}{k}$ , every k-uniform hypergraph on n vertices without t disjoint edges has at most max ${\binom{kt-1}{k}, \binom{n}{k}-\binom{n-t+1}{k}\}$ edges. Although this appears to be a basic instance of the hypergraph Turan problem (with a t-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all $t . This improves upon the best previously known range $t = O\bigl(\frac{n}{k^3}\bigr)$ , which dates back to the 1970s.
106 citations
TL;DR: The authors give upper bounds for the size of 3-uniform hypergraphs avoiding a given odd cycle using the definition of a cycle due to Berge, and extend this bound to m- uniform hyper graphs (for all m ≥ 3), as well as m-un uniform hypergraphS avoiding a cycle of length 2k.
Abstract: Recently, the authors gave upper bounds for the size of 3-uniform hypergraphs avoiding a given odd cycle using the definition of a cycle due to Berge. In the present paper we extend this bound to m-uniform hypergraphs (for all m ⥠3), as well as m-uniform hypergraphs avoiding a cycle of length 2k. Finally we consider non-uniform hypergraphs avoiding cycles of length 2k or 2k + 1. In both cases we can bound |h| by O(n1+1/k) under the assumption that all h ∈ e( ) satisfy |h| ⥠4k2.
95 citations
69 citations
TL;DR: In this paper, the authors studied the percolation phase transition on a random graph G, i.e., the emergence as p increases of a unique giant component in the random subgraph G[p] obtained by keeping edges independently with probability p.
Abstract: Let G = G(d) be a random graph with a given degree sequence d, such as a random r-regular graph where r ⥠3 is fixed and n = |G| â ∞. We study the percolation phase transition on such graphs G, i.e., the emergence as p increases of a unique giant component in the random subgraph G[p] obtained by keeping edges independently with probability p. More generally, we study the emergence of a giant component in G(d) itself as d varies. We show that a single method can be used to prove very precise results below, inside and above the 'scaling window' of the phase transition, matching many of the known results for the much simpler model G(n, p). This method is a natural extension of that used by Bollobas and the author to study G(n, p), itself based on work of Aldous and of Nachmias and Peres; the calculations are significantly more involved in the present setting.
56 citations
TL;DR: In this paper, the authors introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest, and the question then arises whether stable instances of NP-hard problems are easier to solve.
Abstract: We introduce the notion of a stable instance for a discrete optimization problem, and argue that in many practical situations only sufficiently stable instances are of interest. The question then arises whether stable instances of NP-hard problems are easier to solve, and in particular, whether there exist algorithms that solve in polynomial time all sufficiently stable instances of some NP-hard problem. The paper focuses on the Max-Cut problem, for which we show that this is indeed the case.
54 citations
TL;DR: LeSaulnier, Stocker, Wenger and West showed that a rainbow matching of size ⌊k/2⌋ is guaranteed to exist, and Li and Xu proved the conjecture for all other properly coloured complete graphs.
Abstract: A rainbow subgraph of an edge-coloured graph is a subgraph whose edges have distinct colours. The colour degree of a vertex v is the number of different colours on edges incident with v. Wang and Li conjectured that for k ⥠4, every edge-coloured graph with minimum colour degree k contains a rainbow matching of size at least âk/2â. A properly edge-coloured K4 has no such matching, which motivates the restriction k ⥠4, but Li and Xu proved the conjecture for all other properly coloured complete graphs. LeSaulnier, Stocker, Wenger and West showed that a rainbow matching of size âk/2â is guaranteed to exist, and they proved several sufficient conditions for a matching of size âk/2â. We prove the conjecture in full.
50 citations
TL;DR: In this article, it was shown that the Ramsey number of a path does not change if the graph to be coloured is not complete but has a large minimum degree, which is a weaker version of Schelp's conjecture.
Abstract: R. H. Schelp conjectured that if G is a graph with |V(G)| = R(Pn, Pn) such that I´(G) > $$\frac{3|V(G)|}{ 4}$ , then in every 2-colouring of the edges of G there is a monochromatic Pn. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree.
Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching-matching Ramsey number satisfying R(Sn, nK2, nK2) = 3n-1. This extends R(nK2, nK2) = 3n-1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp's conjecture in an asymptotic sense through a standard application of the Regularity Lemma.
It is sad that we are unable to hear Dick Schelp's reaction to our work generated by his conjecture.
48 citations
TL;DR: It is shown that, for all δ>0 and n>n0(δ), if G is a 2-edge coloured graph of order n with δ(G) ≥ 3n/4, then one colour class either contains a monochromatic cycle of length at least (2/3+δ/2)n, or contains monochromaatic cycles of all lengths ℓ ∈ [3, ( 2/3−δ)n
Abstract: Li, Nikiforov and Schelp [13] conjectured that any 2-edge coloured graph G with order n and minimum degree I´(G) > 3n/4 contains a monochromatic cycle of length â, for all â ∈ [4, ân/2â]. We prove this conjecture for sufficiently large n and also find all 2-edge coloured graphs with I´(G)=3n/4 that do not contain all such cycles. Finally, we show that, for all I´>0 and n>n0(I´), if G is a 2-edge coloured graph of order n with I´(G) ⥠3n/4, then one colour class either contains a monochromatic cycle of length at least (2/3+I´/2)n, or contains monochromatic cycles of all lengths â ∈ [3, (2/3-I´)n].
39 citations
TL;DR: It is proved that for p(n) ≥ polylog(n)/n the board G ~ n,p is typically such that Maker can win these Maker–Breaker games asymptotically as fast as possible, i.e., within n+o(n), n/2+o (n) and kn/2-o( n) moves respectively.
Abstract: In this paper we analyse classical Maker-Breaker games played on the edge set of a sparse random board G ~ n,p. We consider the Hamiltonicity game, the perfect matching game and the k-connectivity game. We prove that for p(n) ⥠polylog(n)/n the board G ~ n,p is typically such that Maker can win these games asymptotically as fast as possible, i.e., within n+o(n), n/2+o(n) and kn/2+o(n) moves respectively.
29 citations
TL;DR: A new graph polynomial is introduced that encodes interesting properties of graphs, for example, the number of matchings, thenumber of perfect matching, and, for bipartite graphs, theNumber of independent sets (#BIS).
Abstract: We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings, the number of perfect matchings, and, for bipartite graphs, the number of independent sets (#BIS).
We analyse the complexity of exact evaluation of the polynomial at rational points and show a dichotomy result: for most points exact evaluation is #P-hard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial.
TL;DR: In this paper, it was shown that dense graphs that are hard to make triangle-free have a large packing of pairwise edge-disjoint triangles, where β is the density of the graph and c ⥠is an absolute constant.
Abstract: It is well known that a graph with m edges can be made triangle-free by removing (slightly less than) m/2 edges. On the other hand, there are many classes of graphs which are hard to make triangle-free, in the sense that it is necessary to remove roughly m/2 edges in order to eliminate all triangles.
We prove that dense graphs that are hard to make triangle-free have a large packing of pairwise edge-disjoint triangles. In particular, they have more than m(1/4+cβ) pairwise edge-disjoint triangles where β is the density of the graph and c ⥠is an absolute constant. This improves upon a previous m(1/4-o(1)) bound which follows from the asymptotic validity of Tuza's conjecture for dense graphs. We conjecture that such graphs have an asymptotically optimal triangle packing of size m(1/3-o(1)).
We extend our result from triangles to larger cliques and odd cycles.
TL;DR: It is proved that every connected and locally connected graph with more than 3 vertices contains a homeomorphically irreducible spanning tree (HIST), which confirms the following conjecture due to Archdeacon: every graph that triangulates some surface has a HIST.
Abstract: A spanning tree T of a graph G is called a homeomorphically irreducible spanning tree (HIST) if T does not contain vertices of degree 2. A graph G is called locally connected if, for every vertex v ∈ V(G), the subgraph induced by the neighbourhood of v is connected. In this paper, we prove that every connected and locally connected graph with more than 3 vertices contains a HIST. Consequently, we confirm the following conjecture due to Archdeacon: every graph that triangulates some surface has a HIST, which was proposed as a question by Albertson, Berman, Hutchinson and Thomassen.
TL;DR: This paper provides lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable and shows that this is not necessarily the case.
Abstract: A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by ICF( ) Pach and Tardos proved that for an (2r-1)-uniform hypergraph with m edges, ICF( ) is at most of the order of rm1/r log m, for fixed r and large m They also raised the question whether a similar upper bound holds for r-uniform hypergraphs In this paper we show that this is not necessarily the case Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable
TL;DR: It is shown that for any bridge-addable, monotone class whose elements have vertex set {1,.
Abstract: A class of labelled graphs is bridge-addable if, for all graphs G in and all vertices u and v in distinct connected components of G, the graph obtained by adding an edge between u and v is also in ; the class is monotone if, for all G ∈ and all subgraphs H of G, we have H ∈ . We show that for any bridge-addable, monotone class whose elements have vertex set {1,.i¾ .i¾ .,n}, the probability that a graph chosen uniformly at random from is connected is at least (1-on(1))e-i¾½, where on(1) â 0 as n â ∞. This establishes the special case of the conjecture of McDiarmid, Steger and Welsh when the condition of monotonicity is added. This result has also been obtained independently by Kang and Panagiotou.
TL;DR: In this article, the authors studied the cop number of geometric graphs and showed that with high probability (w.h.p) the number of cops required to catch a robber in finite time is 9.
Abstract: Cops and robbers is a turn-based pursuit game played on a graph G. One robber is pursued by a set of cops. In each round, these agents move between vertices along the edges of the graph. The cop number c(G) denotes the minimum number of cops required to catch the robber in finite time. We study the cop number of geometric graphs. For points x1,.i¾ .i¾ ., xn ∈ â2, and r ∈ â+, the vertex set of the geometric graph G(x1,.i¾ .i¾ ., xn; r) is the graph on these n points, with xi, xj adjacent when â¥xi-xj⥠⤠r. We prove that c(G) ⤠9 for any connected geometric graph G in â2 and we give an example of a connected geometric graph with c(G) = 3. We improve on our upper bound for random geometric graphs that are sufficiently dense. Let (n,r) denote the probability space of geometric graphs with n vertices chosen uniformly and independently from [0,1]2. For G ∈ (n,r), we show that with high probability (w.h.p.), if r ⥠K1 (log n/n)1/4 then c(G) ⤠2, and if r ⥠K2(log n/n)1/5 then c(G) = 1, where K1, K2 > 0 are absolute constants. Finally, we provide a lower bound near the connectivity regime of (n,r): if r ⤠K3 log n/ then c(G) > 1 w.h.p., where K3 > 0 is an absolute constant.
TL;DR: Several results about edge-colourings of bounded-degree graphs that are related to degree Ramsey numbers of paths are proved, including the stronger statement that every graph with maximum degree at most 4 has a 2-edge-colouring such that the subgraph in each colour class has girth at least 5.
Abstract: Let H G mean that every s-colouring of E(H) produces a monochromatic copy of G in some colour class. Let the s-colour degree Ramsey number of a graph G, written RΔ(G; s), be min{Δ(H): H G}. If T is a tree in which one vertex has degree at most k and all others have degree at most âk/2â, then RΔ(T; s) = s(k-1) + Iµ, where Iµ = 1 when k is odd and Iµ = 0 when k is even. For general trees, RΔ(T; s) ⤠2s(Δ(T)-1).
To study sharpness of the upper bound, consider the double-starSa,b, the tree whose two non-leaf vertices have degrees a and b. If a ⤠b, then RΔ(Sa,b; 2) is 2b-2 when a
TL;DR: The exact maximum is determined in this paper for some small i.e. families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximumsize of the intersecting family only by one.
Abstract: Let be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality | | ⤠2n-1. Suppose that | |=2n-1 + i. Choose the members of independently with probability p (delete them with probability 1-p). The new family is intersecting with a certain probability. We try to maximize this probability by choosing appropriately. The exact maximum is determined in this paper for some small i. The analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all $\lfloor \frac{n}{ 2}\rfloor$ -element subsets. We determine the most probably inclusion-free family too, when the number of members is $\binom{n}{ \lfloor \frac{n}{ 2}\rfloor} +1$ .
TL;DR: In this article, the Freiman-Ruzsa theorem was shown to be poly(K)4K/4K when both A and B are unions of cosets of a certain subgroup of â¤2n.
Abstract: Let A and B be two affinely generating sets of â¤2n. As usual, we denote their Minkowski sum by A+B. How small can A+B be, given the cardinalities of A and B? We give a tight answer to this question. Our bound is attained when both A and B are unions of cosets of a certain subgroup of â¤2n. These cosets are arranged as Hamming balls, the smaller of which has radius 1.
By similar methods, we re-prove the Freiman-Ruzsa theorem in â¤2n, with an optimal upper bound. Denote by F(K) the maximal spanning constant |a A a|/|A| over all subsets A â â¤2n with doubling constant |A+A|/|A| ⤠K. We explicitly calculate F(K), and in particular show that 4K/4K ⤠F(K)â (1+o(1)) ⤠4K/2K. This improves the estimate F(K) = poly(K)4K, found recently by Green and Tao [17] and by Konyagin [23].
TL;DR: This paper shows that the families [n](≥r) = {A ⊂ [n]: |A| ≥ r} are optimal for the appropriate values of i, thereby proving the conjecture for this sequence of values.
Abstract: A family of sets is said to be intersecting if A â© B â â for all A, B ∈ . It is a well-known and simple fact that an intersecting family of subsets of [n] = {1, 2,.i¾ .i¾ ., n} can contain at most 2n-1 sets. Katona, Katona and Katona ask the following question. Suppose instead â [n] satisfies | | = 2n-1 + i for some fixed i > 0. Create a new family p by choosing each member of independently with some fixed probability p. How do we choose to maximize the probability that p is intersecting? They conjecture that there is a nested sequence of optimal families for i = 1, 2,.i¾ .i¾ ., 2n-1. In this paper, we show that the families [n](â¥r) = {A â [n]: |A| ⥠r} are optimal for the appropriate values of i, thereby proving the conjecture for this sequence of values. Moreover, we show that for intermediate values of i there exist optimal families lying between those we have found. It turns out that the optimal families we find simultaneously maximize the number of intersecting subfamilies of each possible order.
Standard compression techniques appear inadequate to solve the problem as they do not preserve intersection properties of subfamilies. Instead, our main tool is a novel compression method, together with a way of 'compressing subfamilies', which may be of independent interest.
TL;DR: It is proved that the threshold for the appearance of a k-regular subgraph in Gn,p is at most the threshold of a non-empty (k+1)-core.
Abstract: We prove that the threshold for the appearance of a k-regular subgraph in Gn,p is at most the threshold for the appearance of a non-empty (k+1)-core. This improves a result of Pralat, Verstraete and Wormald [5] and proves a conjecture of Bollobas, Kim and Verstraete [3].
TL;DR: It is shown, by integrating multiple biophysical tools, that the majority of IDPs that regulate PP1, prefer a conformational selection model.
Abstract: Intrinsically disordered but biologically active proteins, commonly referred to as IDPs, are readily identified in many biological systems and play critical roles in multiple protein regulatory processes. While disordered in their unbound states, IDPs often, but not always, fold upon binding with their protein interaction partners. Here, we discuss how a class of IDPs directs the targeting, specificity and activity of Protein Phosphatase 1 (PP1). PP1 is major ser/thr phosphatase that plays a critical role in a broad range of biological processes, from muscle contraction to memory formation. In the cell, PP1 is regulated through its interaction with more than 200 regulatory proteins, the majority of which are IDPs. Critically, these PP1:regulatory protein holoenzyme complexes confer specificity to PP1 and are thus the functional forms of the PP1 enzyme in vivo. Furthermore, we discuss the distinct modes of interaction utilized by IDPs to complex with their protein binding partners. We subsequently show, by integrating multiple biophysical tools, that the majority of IDPs that regulate PP1, prefer a conformational selection model.
TL;DR: A variant of the crossing number for an embedding of a graph G into ℝ3 is defined, and a lower bound on it is proved which almost implies the classical crossing lemma.
Abstract: We define a variant of the crossing number for an embedding of a graph G into â3, and prove a lower bound on it which almost implies the classical crossing lemma. We also give sharp bounds on the rectilinear space crossing numbers of pseudo-random graphs.
TL;DR: This work determines the smallest interval such that if p > pmax(G), then every 1-independent bond percolation model on G with bond probability p percolates, and for p < pmin(G) none does.
Abstract: Given a locally finite connected infinite graph G, let the interval [pmin(G), pmax(G)] be the smallest interval such that if p >pmax(G), then every 1-independent bond percolation model on G with bond probability p percolates, and for p
TL;DR: It is shown that if b ≥ 12, then fb(n) is established for some absolute constant B, thus establishing fb (n) up to polylogarithmic factors, which leaves open the interesting case b = 2, which is the case of 2-regular subgraphs.
Abstract: A subgraph of a hypergraph H is even if all its degrees are positive even integers, and b-bounded if it has maximum degree at most b. Let fb(n) denote the maximum number of edges in a linearn-vertex 3-uniform hypergraph which does not contain a b-bounded even subgraph. In this paper, we show that if b ⥠12, then \[ \frac{n \log n}{3 b\log \log n} \leq f_b(n) \leq Bn(\log n)^2 \] for some absolute constant B, thus establishing fb(n) up to polylogarithmic factors. This leaves open the interesting case b = 2, which is the case of 2-regular subgraphs. We are able to show for some constants c, C > 0 that \[ c n\log n \leq f_2(n) \leq Cn^{3/2}(\log n)^5. \] We conjecture that f2(n) = n1 + o(1) as n â ∞.
TL;DR: It is verified that Scott's conjecture that the class of graphs with no induced subdivision of a given graph is χ-bounded for maximal triangle-free graphs is verified.
Abstract: Scott conjectured in [6] that the class of graphs with no induced subdivision of a given graph is I-bounded. We verify his conjecture for maximal triangle-free graphs.