Authors:Adam Sheffer; Endre Szabó; Joshua Zahl 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: 2017-02-13 DOI: 10.1007/s00493-016-3441-7

Authors:Eli Berger Abstract: We prove that in any graph containing no subdivision of an infinite clique there exists a partition of the vertices into two parts, satisfying the condition that every vertex has at least as many neighbors in the part not containing it as it has in its own part. PubDate: 2017-02-13 DOI: 10.1007/s00493-015-3261-1

Authors:Aart Blokhuis; Hao Chen Abstract: A set U of unit vectors is selectively balancing if one can find two disjoint subsets U + and U -, not both empty, such that the Euclidean distance between the sum of U + and the sum of U - is smaller than 1. We prove that the minimum number of unit vectors that guarantee a selectively balancing set in R n is asymptotically 1/2nlogn. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3635-z

Authors:Brandon Hanson Abstract: Guth and Katz proved that any point set P in the plane determines Ω( P /log P ) distinct distances. We show that when near to this lower bound, a point set P of the form A × A must satisfy A-A ≪ A-A 2-1/8. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3665-6

Authors:Reinhard Diestel; Malte Müller 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: 2017-02-13 DOI: 10.1007/s00493-016-3516-5

Authors:Javier Cilleruelo; Rafael Tesoro Abstract: We study extremal problems about sets of integers that do not contain sumsets with summands of prescribed size. We analyse both finite sets and infinite sequences. We also study the connections of these problems with extremal problems of graphs and hypergraphs. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3444-4

Authors:Daniele Bartoli; Massimo Giulietti; Giuseppe Marino; Olga Polverino 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: 2017-02-13 DOI: 10.1007/s00493-016-3531-6

Authors:João Araújo; Wolfram Bentz; Edward Dobson; Janusz Konieczny; Joy Morris Abstract: We characterize the automorphism groups of circulant digraphs whose connection sets are relatively small, and of unit circulant digraphs. For each class, we either explicitly determine the automorphism group or we show that the graph is a “normal” circulant, so the automorphism group is contained in the normalizer of a cycle. Then we use these characterizations to prove results on the automorphisms of the endomorphism monoids of those digraphs. The paper ends with a list of open problems on graphs, number theory, groups and semigroups. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3403-0

Authors:Tristram C. Bogart; Edward D. Kim Abstract: We construct a sequence of subset partition graphs satisfying the dimension reduction, adjacency, strong adjacency, and endpoint count properties whose diameter has a superlinear asymptotic lower bound. These abstractions of polytope graphs give further evidence against the Linear Hirsch Conjecture. PubDate: 2017-02-13 DOI: 10.1007/s00493-016-3327-8

Authors:Nick Gill; Neil I. Gillespie; Jason Semeraro 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 M 13 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 M 13 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: 2017-02-13 DOI: 10.1007/s00493-016-3433-7

Authors:David Bevan 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: 2017-02-13 DOI: 10.1007/s00493-016-3349-2

Authors:John Bamberg; Melissa Lee; Koji Momihara; Qing Xiang Abstract: In this paper, we construct an infinite family of hemisystems of the Hermitian surface H(3, q 2). In particular, we show that for every odd prime power q congruent to 3 modulo 4, there exists a hemisystem of H(3, q 2) admitting \(C_{\left( {q^3 + 1} \right)/4} :C_3 \) . PubDate: 2017-01-10 DOI: 10.1007/s00493-016-3525-4

Authors:Gabe Cunningham Abstract: A chiral polyhedron with Schläfli symbol {p,q} is called tight if it has 2pq flags, which is the minimum possible. In this paper, we fully characterize the Schläfli symbols of tight chiral polyhedra. We also provide presentations for the automorphism groups of several families of tight chiral polyhedra. PubDate: 2017-01-10 DOI: 10.1007/s00493-016-3505-8

Authors:Rohan Kapadia Abstract: For a set of matroids M, let ex M (n) be the maximum size of a simple rank-n matroid in M. We prove that, for any finite field F, if M is a minor-closed class of F-representable matroids of bounded branch-width, then lim n→ ∞ ex M (n)/n exists and is a rational number, ∆. We also show that ex M (n) - ∆n is periodic when n is sufficiently large and that ex M is achieved by a subclass of M of bounded path-width. PubDate: 2017-01-10 DOI: 10.1007/s00493-016-3425-7

Authors:Misha Rudnev Abstract: We prove an incidence theorem for points and planes in the projective space P3 over any Field F, whose characteristic p ≠ 2. An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving p if p > 0. This yields a bound on the number of incidences between m points and n planes in ℙ3, with m≥n as $$O\left( {m\sqrt n + mk} \right)$$ , where k is the maximum number of collinear planes, provided that n = O(p 2) if p > 0. Examples show that this bound cannot be improved without additional assumptions. This gives one a vehicle to establish geometric incidence estimates when p >0. For a non-collinear point set S⊆F2 and a non-degenerate symmetric or skew-symmetric bilinear form ω, the number of distinct values of ω on pairs of points of S is \(\Omega \left[ {\min \left( {{{\left S \right }^{\frac{2}{3}}},p} \right)} \right]\) . This is also the best known bound over ℝ, where it follows from the Szemerédi-Trotter theorem. Also, a set S ⊆ F3, not supported in a single semi-isotropic plane contains a point, from which \(\Omega \left[ {\min \left( {{{\left S \right }^{\frac{1}{2}}},p} \right)} \right]\) distinct distances to other points of S are attained. PubDate: 2017-01-10 DOI: 10.1007/s00493-016-3329-6

Authors:Bence Csajbók Abstract: If B is a minimal blocking set of size less than 3(q+1)=2 in PG(2,q), q is a power of the prime p, then Szőnyi’s result states that each line meets B in 1 (mod p) points. It follows that B cannot have bisecants, i.e., lines meeting B in exactly two points. If q >13, then there is only one known minimal blocking set of size 3(q+1)=2 in PG(2, q), the so-called projective triangle. This blocking set is of Rédei type and it has 3(q-1)=2 bisecants, which have a very strict structure. We use polynomial techniques to derive structural results on Rédei type blocking sets from information on their bisecants. We apply our results to point sets of PG(2, q) with few odd-secants. In particular, we improve the lower bound of Balister, Bollobás, Füredi and Thompson on the number of odd-secants of a (q+2)-set in PG(2, q) and we answer a related open question of Vandendriessche. We prove structural results for semiovals and derive the non existence of semiovals of size q+3 when p≠3 and q>5. This extends a result of Blokhuis who classified semiovals of size q+2, and a result of Bartoli who classified semiovals of size q+3 when q ≤ 17. In the q even case we can say more applying a result of Szőnyi and Weiner about the stability of sets of even type. We also obtain a new proof to a result of Gács and Weiner about (q+t, t)-arcs of type (0, 2, t) and to one part of a result of Ball, Blokhuis, Brouwer, Storme and Szőnyi about functions over GF(q) determining less than (q+3)/2 directions. PubDate: 2017-01-10 DOI: 10.1007/s00493-016-3442-6

Authors:Jim Geelen; Rohan Kapadia Abstract: We give polynomial-time randomized algorithms for computing the girth and the cogirth of binary matroids that are low-rank perturbations of graphic matroids. PubDate: 2017-01-10 DOI: 10.1007/s00493-016-3445-3

Authors:Genghua Fan Abstract: Let G be a bridgeless graph and denote by cc(G) the shortest length of a cycle cover of G. Let V 2(G) be the set of vertices of degree 2 in G. It is known that if cc(G)≤1.4 E(G) for every bridgeless graph G with V 2(G) ≤1/10 E(G) , then the Cycle Double Cover Conjecture is true. The best known result cc(G)≤5/3 E(G) (≈1.6667 E(G) ) was established over 30 years ago. Recently, it was proved that cc(G) ≤ 44/27 E(G) (≈ 1.6296 E(G) ) for loopless graphs with minimum degree at least 3. In this paper, we obtain results on integer 4-flows, which are used to find bounds for cc(G). We prove that if G has minimum degree at least 3 (loops being allowed), then cc(G)<1.6258 E(G) . As a corollary, adding loops to vertices of degree 2, we obtain that cc(G)<1.6466 E(G) for every bridgeless graph G with V 2(G) ≤1/30 E(G) . PubDate: 2016-12-22 DOI: 10.1007/s00493-016-3379-9

Authors:Peter Allen; Julia Böttcher; Hiệp Hàn; Yoshiharu Kohayakawa; Yury Person Abstract: We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph G is (ε,p,k,ℓ)-pseudorandom if for all disjoint X and Y ⊂ V(G) with X ≥εp k n and Y ≥εp ℓ n we have e(X,Y)=(1±ε)p X Y . We prove that for all β>0 there is an ε>0 such that an (ε,p,1,2)-pseudorandom graph on n vertices with minimum degree at least βpn contains the square of a Hamilton cycle. In particular, this implies that (n,d,λ)-graphs with λ≪d 5/2 n -3/2 contain the square of a Hamilton cycle, and thus a triangle factor if n is a multiple of 3. This improves on a result of Krivelevich, Sudakov and Szabó [27]. We also extend our result to higher powers of Hamilton cycles and establish corresponding counting versions. PubDate: 2016-12-22 DOI: 10.1007/s00493-015-3228-2