Authors:Daniele Bartoli; Massimo Giulietti; Giuseppe Marino; Olga Polverino Pages: 255 - 278 Abstract: Explicit constructions of infinite families of scattered F q -linear sets in PG(r-1, q t ) of maximal rank rt/2, for t ≥ 4 even, are provided. When q = 2, these linear sets correspond to complete caps in AG(r,2 t ) fixed by a translation group of size 2rt/2. The doubling construction applied to such caps gives complete caps in AG(r+1, 2 t ) of size 2rt/2+1. For Galois spaces of even dimension greater than 2 and even square order, this solves the long-standing problem of establishing whether the theoretical lower bound for the size of a complete cap is substantially sharp. PubDate: 2018-04-01 DOI: 10.1007/s00493-016-3531-6 Issue No:Vol. 38, No. 2 (2018)

Authors:David Bevan Pages: 279 - 303 Abstract: We prove that the set of growth rates of permutation classes includes an infinite sequence of intervals whose infimum is θ B ≈ 2:35526, and that it also contains every value at least θ B ≈ 2:35698. These results improve on a theorem of Vatter, who determined that there are permutation classes of every growth rate at least λ A ≈ 2:48187. Thus, we also refute his conjecture that the set of growth rates below λ A is nowhere dense. Our proof is based upon an analysis of expansions of real numbers in non-integer bases, the study of which was initiated by Rényi in the 1950s. In particular, we prove two generalisations of a result of Pedicini concerning expansions in which the digits are drawn from sets of allowed values. PubDate: 2018-04-01 DOI: 10.1007/s00493-016-3349-2 Issue No:Vol. 38, No. 2 (2018)

Authors:Amin Coja-Oghlan; Charilaos Efthymiou; Nor Jaafari Pages: 341 - 380 Abstract: Let G = G(n, m) be a random graph whose average degree d = 2m/n is below the k-colorability threshold. If we sample a k-coloring σ of G uniformly at random, what can we say about the correlations between the colors assigned to vertices that are far apart' According to a prediction from statistical physics, for average degrees below the so-called condensation threshold dk,cond, the colors assigned to far away vertices are asymptotically independent [Krzakala et al.: Proc. National Academy of Sciences 2007]. We prove this conjecture for k exceeding a certain constant k0. More generally, we investigate the joint distribution of the k-colorings that σ induces locally on the bounded-depth neighborhoods of any fixed number of vertices. In addition, we point out an implication on the reconstruction problem. PubDate: 2018-04-01 DOI: 10.1007/s00493-016-3394-x Issue No:Vol. 38, No. 2 (2018)

Authors:Reinhard Diestel; Malte Müller Pages: 381 - 398 Abstract: The connected tree-width of a graph is the minimum width of a tree-decomposition whose parts induce connected subgraphs. Long cycles are examples of graphs that have small tree-width but large connected tree-width. We show that a graph has small connected tree-width if and only if it has small tree-width and contains no long geodesic cycle. We further prove a connected analogue of the duality theorem for tree-width: a finite graph has small connected tree-width if and only if it has no bramble whose connected covers are all large. Both these results are qualitative: the bounds are good but not tight. We show that graphs of connected tree-width k are k-hyperbolic, which is tight, and that graphs of tree-width k whose geodesic cycles all have length at most ℓ are ⌊3/2l(k-1)⌋-hyperbolic. The existence of such a function h(k, ℓ) had been conjectured by Sullivan. PubDate: 2018-04-01 DOI: 10.1007/s00493-016-3516-5 Issue No:Vol. 38, No. 2 (2018)

Authors:Nick Gill; Neil I. Gillespie; Jason Semeraro Pages: 399 - 442 Abstract: To each supersimple 2-(n,4,λ) design D one associates a ‘Conway groupoid’, which may be thought of as a natural generalisation of Conway’s Mathieu groupoid M13 which is constructed from P3. We show that Sp2m(2) and 22m. Sp2m(2) naturally occur as Conway groupoids associated to certain designs. It is shown that the incidence matrix associated to one of these designs generates a new family of completely transitive F2-linear codes with minimum distance 4 and covering radius 3, whereas the incidence matrix of the other design gives an alternative construction of a previously known family of completely transitive codes. We also give a new characterization of M13 and prove that, for a fixed λ > 0; there are finitely many Conway groupoids for which the set of morphisms does not contain all elements of the full alternating group. PubDate: 2018-04-01 DOI: 10.1007/s00493-016-3433-7 Issue No:Vol. 38, No. 2 (2018)

Authors:Adam Sheffer; Endre Szabó; Joshua Zahl Pages: 487 - 499 Abstract: We prove an incidence theorem for points and curves in the complex plane. Given a set of m points in ℝ2 and a set of n curves with k degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is \(O\left( {{m^{\frac{k}{{2k - 1}}}}{n^{\frac{{2k - 2}}{{2k - 1}}}} + m + n} \right)\) . We establish the slightly weaker bound \({O_\varepsilon }\left( {{m^{\frac{k}{{2k - 1}} + \varepsilon }}{n^{\frac{{2k - 2}}{{2k - 1}}}} + m + n} \right)\) on the number of incidences between m points and n (complex) algebraic curves in ℂ2 with k degrees of freedom. We combine tools from algebraic geometry and differential geometry to prove a key technical lemma that controls the number of complex curves that can be contained inside a real hypersurface. This lemma may be of independent interest to other researchers proving incidence theorems over ℂ. PubDate: 2018-04-01 DOI: 10.1007/s00493-016-3441-7 Issue No:Vol. 38, No. 2 (2018)

Authors:Jacob Fox; János Pach; Andrew Suk Pages: 501 - 509 Abstract: We establish the following result related to Erdős’s problem on distinct distances. Let V be an n-element planar point set such that any p members of V determine at least \(\left( {\begin{array}{*{20}{c}} p \\ 2 \end{array}} \right) - p + 6\) distinct distances. Then V determines at least \(n^{\tfrac{8} {7} - o(1)}\) distinct distances, as n tends to infinity. PubDate: 2018-04-01 DOI: 10.1007/s00493-016-3637-x Issue No:Vol. 38, No. 2 (2018)

Authors:Yu Yokoi Abstract: In 1995, Galvin proved that a bipartite graph G admits a list edge coloring if every edge is assigned a color list of length Δ(G) the maximum degree of the graph. This result was improved by Borodin, Kostochka and Woodall, who proved that G still admits a list edge coloring if every edge e=st is assigned a list of max{d G (s);d G (t)} colors. Recently, Iwata and Yokoi provided the list supermodular coloring theorem that extends Galvin's result to the setting of Schrijver's supermodular coloring. This paper provides a common generalization of these two extensions of Galvin's result. PubDate: 2018-05-17 DOI: 10.1007/s00493-018-3830-1

Authors:Julien Bensmail; Ararat Harutyunyan; Tien-Nam Le; Stéphan Thomasse Abstract: In 2006, Barát and Thomassen conjectured that there is a function f such that, for every fixed tree T with t edges, every f(t)-edge-connected graph with its number of edges divisible by t has a partition of its edges into copies of T. This conjecture was recently verified by the current authors and Merker [1]. We here further focus on the path case of the Barát-Thomassen conjecture. Before the aforementioned general proof was announced, several successive steps towards the path case of the conjecture were made, notably by Thomassen [11,12,13], until this particular case was totally solved by Botler, Mota, Oshiro andWakabayashi [2]. Our goal in this paper is to propose an alternative proof of the path case with a weaker hypothesis: Namely, we prove that there is a function f such that every 24-edge-connected graph with minimum degree f(t) has an edge-partition into paths of length t whenever t divides the number of edges. We also show that 24 can be dropped to 4 when the graph is eulerian. PubDate: 2018-05-17 DOI: 10.1007/s00493-017-3661-5

Authors:Henning Bruhn; Matthias Heinlein; Felix Joos Abstract: We prove that the set of long cycles has the edge-Erdős-Pósa property: for every fixed integer ℓ ≥ 3 and every k ∈ ℕ, every graph G either contains k edge-disjoint cycles of length at least ℓ (long cycles) or an edge set X of size O(k2 logk+kℓ) such that G—X does not contain any long cycle. This answers a question of Birmelé, Bondy, and Reed (Combinatorica 27 (2007), 135-145). PubDate: 2018-05-17 DOI: 10.1007/s00493-017-3669-x

Authors:Sang-Il Oum; Sounggun Wee Abstract: O. Einstein (2008) proved Bollobás-type theorems on intersecting families of ordered sets of finite sets and subspaces. Unfortunately, we report that the proof of a theorem on ordered sets of subspaces had a mistake. We prove two weaker variants. PubDate: 2018-04-17 DOI: 10.1007/s00493-018-3812-3

Authors:Tamás Fleiner Abstract: We prove an extension of Galvin’s theorem, namely that any graph is L-edge-choosable if L(e) ≥χ′(G) and the edge-lists of no odd cycle contain a common colour. PubDate: 2018-04-17 DOI: 10.1007/s00493-018-3888-9

Authors:Choongbum Lee; Brandon Tran Abstract: A simple graph-product type construction shows that for all natural numbers r≥q, there exists an edge-coloring of the complete graph on 2 r vertices using r colors where the graph consisting of the union of any q color classes has chromatic number 2 q . We show that for each fixed natural number q, if there exists an edge-coloring of the complete graph on n vertices using r colors where the graph consisting of the union of any q color classes has chromatic number at most 2 q − 1, then n must be sub-exponential in r. This answers a question of Conlon, Fox, Lee, and Sudakov. PubDate: 2018-04-17 DOI: 10.1007/s00493-017-3474-6

Authors:Carlos Hoppen; Nicholas Wormald Abstract: We introduce a general class of algorithms and analyse their application to regular graphs of large girth. In particular, we can transfer several results proved for random regular graphs into (deterministic) results about all regular graphs with sufficiently large girth. This reverses the usual direction, which is from the deterministic setting to the random one. In particular, this approach enables, for the first time, the achievement of results equivalent to those obtained on random regular graphs by a powerful class of algorithms which contain prioritised actions. As a result, we obtain new upper or lower bounds on the size of maximum independent sets, minimum dominating sets, maximum k-independent sets, minimum k-dominating sets and maximum k-separated matchings in r-regular graphs with large girth. PubDate: 2018-04-17 DOI: 10.1007/s00493-016-3236-x

Authors:Sean Eberhard; Stefan-Christoph Virchow Abstract: We consider the probability p(S n ) that a pair of random permutations generates either the alternating group A n or the symmetric group S n . Dixon (1969) proved that p(S n ) approaches 1 as n→∞ and conjectured that p(S n ) = 1 − 1/n+o(1/n). This conjecture was verified by Babai (1989), using the Classification of Finite Simple Groups. We give an elementary proof of this result; specifically we show that p(S n ) = 1 − 1/n+O(n−2+ε). Our proof is based on character theory and character estimates, including recent work by Schlage-Puchta (2012). PubDate: 2018-03-27 DOI: 10.1007/s00493-017-3629-5

Authors:Nathan Bowler; Johannes Carmesin; Robin Christian Abstract: We introduce a class of infinite graphic matroids that contains all the motivating examples and satisfies an extension of Tutte’s excluded minors characterisation of finite graphic matroids.We prove that its members can be represented by certain ‘graph-like’ topological spaces previously considered by Thomassen and Vella. PubDate: 2018-03-27 DOI: 10.1007/s00493-016-3178-3

Authors:Türker Bıyıkoğlu; Yusuf Civan Abstract: We present new combinatorial results on the calculation of (Castelnuovo-Mumford) regularity of graphs. We introduce the notion of a prime graph over a field k, which we define to be a connected graph with regk(G − x) < regk(G) for any vertex x ∈ V (G). We then exhibit some structural properties of prime graphs. This enables us to provide upper bounds to the regularity involving the induced matching number im(G). We prove that reg(G) ≤ (Γ(G)+1)im(G) holds for any graph G, where Γ(G)=max{ N G [x]\N G [y] : xy ∈ E(G)} is the maximum privacy degree of G and N G [x] is the closed neighbourhood of x in G. In the case of claw-free graphs, we verify that this bound can be strengthened by showing that reg(G)≤2im(G). By analysing the effect of Lozin transformations on graphs, we narrow the search for prime graphs into graphs having maximum degree at most three. We show that the regularity of such graphs G is bounded above by 2im(G)+1. Moreover, we prove that any non-trivial Lozin operation preserves the primeness of a graph. That enables us to generate many new prime graphs from the existing ones. We prove that the inequality reg(G/e)≤reg(G)≤reg(G/e)+1 holds for the contraction of any edge e of a graph G. This implies that reg(H) ≤ reg(G) whenever H is an edge contraction minor of G. Finally, we show that there exist connected graphs satisfying reg(G)=n and im(G)=k for any two integers n ≥ k ≥ 1. The proof is based on a result of Januszkiewicz and Świa̦tkowski on the existence of Gromov hyperbolic right angled Coxeter groups of arbitrarily large virtual cohomological dimension, accompanied with Lozin operations. In an opposite direction, we show that if G is a 2K2-free prime graph, then reg(G)≤(δ(G)+3)/2, where δ(G) is the minimum degree of G. PubDate: 2018-03-27 DOI: 10.1007/s00493-017-3450-1

Authors:Ron Holzman; Nitzan Tur Abstract: Lovász proved (see [7]) that given real numbers p1,..., p n , one can round them up or down to integers ϵ1,..., ϵ n , in such a way that the total rounding error over every interval (i.e., sum of consecutive p i ’s) is at most 1-1/n+1. Here we show that the rounding can be done so that for all \(d = 1,...,\left\lfloor {\frac{{n + 1}}{2}} \right\rfloor \) , the total rounding error over every union of d intervals is at most (1- d/n+1) d. This answers a question of Bohman and Holzman [1], who showed that such rounding is possible for each value of d separately. PubDate: 2018-03-05 DOI: 10.1007/s00493-017-3769-7

Authors:Michael Krivelevich Abstract: We provide sufficient conditions for the existence of long cycles in locally expanding graphs, and present applications of our conditions and techniques to Ramsey theory, random graphs and positional games. PubDate: 2018-03-05 DOI: 10.1007/s00493-017-3701-1

Authors:Carsten Lange; Vincent Pilaud Abstract: An associahedron is a polytope whose vertices correspond to triangulations of a convex polygon and whose edges correspond to flips between them. Using labeled polygons, C. Hohlweg and C. Lange constructed various realizations of the associahedron with relevant properties related to the symmetric group and the permutahedron. We introduce the spine of a triangulation as its dual tree together with a labeling and an orientation. This notion extends the classical understanding of the associahedron via binary trees, introduces a new perspective on C. Hohlweg and C. Lange’s construction closer to J.-L. Loday’s original approach, and sheds light upon the combinatorial and geometric properties of the resulting realizations of the associahedron. It also leads to noteworthy proofs which shorten and simplify previous approaches. PubDate: 2018-02-07 DOI: 10.1007/s00493-015-3248-y