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We obtain the cardinality of the lattice of characteristic subspaces of a nilpotent Jordan matrix when the underlying field is G F ( 2 ) , the only field where the lattices of characteristic and hyperinvariant subspaces can be different. If the characteristic polynomial of the matrix splits in the field, the general case can be reduced to the nilpotent Jordan case. Results are complex and highly combinatorial, and include the design of an algorithm. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we consider planar graphs in which each vertex is not incident to some cycles of given lengths, but all vertices can have different restrictions. This generalizes the approach based on forbidden cycles which corresponds to the case where all vertices have the same restrictions on the incident cycles. We prove that a planar graph is 3-choosable if it is satisfied one of the following conditions: [(1)] has no cycles of length 4 or 9 and no 6-cycle is adjacent to a 3-cycle. Moreover, for each vertex , there exists an integer such that is not incident to cycles of length . [(2)] has no cycles of length 4, 7, or 9, and for each vertex , there exists an integer such that is not incident to cycles of length . This result generalizes several previously published results (Zhang and Wu, 2005 , Chen et al., 2008 , Shen and Wang, 2007 , Zhang and Wu, 2004 , Shen et al., 2011 ). [Copyright &y& Elsevier]

Abstract: A disk graph is the intersection graph of disks in the plane, and a unit disk graph is the intersection graph of same radius disks in the plane. We give upper and lower bounds on the number of labelled unit disk and disk graphs on n vertices. We show that the number of disk graphs on n vertices is and the number of unit disk graphs on n vertices is . [Copyright &y& Elsevier]

Abstract: A hexangulation G is a plane graph with each face bounded by a cycle of length six. Define three kinds of transformations A, B and C for hexangulations. It was proved that any two hexangulations with the same order can be transformed into each other by a repeated application of A, B and C, and that without any one of the three transformations, there is a hexangulation which cannot be transformed only by the remaining two transformations. In this article, focusing on the fact that a hexangulation is always bipartite, we consider transformations of hexangulations preserving its bipartition. Let G be any hexangulation with bipartition , where , and we call the bipartition size of G. Observe that both A and C preserve the bipartition size, but B does not. Then we first prove that G satisfies . Secondly, for any hexangulation with the same partition size of G, we prove that if , then G can be transformed into only by A and C, and if , then G can be transformed into only by , where is a transformation preserving the bipartition size and consisting of B twice. [Copyright &y& Elsevier]

Abstract: We consider the following cake cutting game: Alice chooses a set P of n points in the square (cake) , where ; Bob cuts out n axis-parallel rectangles with disjoint interiors, each of them having a point of P as the lower left corner; Alice keeps the rest. It has been conjectured that Bob can always secure at least half of the cake. This remains unsettled, and it is not even known whether Bob can get any positive fraction independent of n. We prove that if Alice can force Bobʼs share to tend to zero, then she must use very many points; namely, to prevent Bob from gaining more than of the cake, she needs at least points. [Copyright &y& Elsevier]

Abstract: The double-sided shadow minimization problem for the Boolean cube layer is considered. It is shown that the intersection of Hamming ball of radius 2 and the layer has the minimal double-sided shadow, and optimal families of a smaller size are described. [Copyright &y& Elsevier]

Abstract: A 3-simplex is a collection of four sets with empty intersection such that any three of them have nonempty intersection. We show that the maximum size of a set system on elements without a -simplex is for all , with equality only achieved by the family of sets containing a given element or of size at most . This extends a result of Keevash and Mubayi, who showed the conclusion for sufficiently large. [Copyright &y& Elsevier]

Abstract: A local algorithm with local horizon is a distributed algorithm that runs in synchronous communication rounds; here is a constant that does not depend on the size of the network. As a consequence, the output of a node in a local algorithm only depends on the input within hops from the node. We give tight bounds on the local horizon for a class of local algorithms for combinatorial problems on unit-disk graphs (UDGs). Most of our bounds are due to a refined analysis of existing approaches, while others are obtained by suggesting new algorithms. The algorithms we consider are based on network decompositions guided by a rectangular tiling of the plane. The algorithms are applied to matching, independent set, graph colouring, vertex cover, and dominating set. We also study local algorithms on quasi-UDGs, which are a popular generalisation of UDGs, aimed at more realistic modelling of communication between the network nodes. Analysing the local algorithms on quasi-UDGs allows one to assume that the nodes know their coordinates only approximately, up to an additive error. Despite the localisation error, the quality of the solution to problems on quasi-UDGs remains the same as for the case of UDGs with perfect location awareness. We analyse the increase in the local horizon that comes along with moving from UDGs to quasi-UDGs. [Copyright &y& Elsevier]

Abstract: Consider the unit circle with distance function measured along the circle. We show that for every selection of points there exists such that . We also discuss a game theoretic interpretation of this result. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study graph embeddings in pseudosurfaces formed by three spheres sharing at most two points each pair, and in such a way that all vertices in the graph are in the same face. Our results and examples show that the behaviour of outer embeddings in these pseudosurfaces is rather different from embeddings on the pseudosurfaces considered in the literature so far. [ABSTRACT FROM AUTHOR]

Abstract: Three-level secret sharing schemes arising from the vector space construction over a finite field are presented. Compared to the previously known schemes, they allow a larger number of participants with respect to the size of . The key tool is the twisted cubic of . [Copyright &y& Elsevier]

Abstract: We consider point–line geometries having three points on every line, having three lines through every point (bislim geometries), and containing triangles. We classify such geometries under the hypothesis of the existence of a collineation group acting transitively on the point set. [Copyright &y& Elsevier]

Abstract: We enumerate all dissections of an equilateral triangle into smaller equilateral triangles up to size 20, where each triangle has integer side lengths. A perfect dissection has no two triangles of the same side, counting up- and down-oriented triangles as different. We computationally prove Tutte’s conjecture that the smallest perfect dissection has size 15 and we find all perfect dissections up to size 20. [Copyright &y& Elsevier]

Abstract: We give a construction of -regular graphs of girth using only geometrical and combinatorial properties that appear in any -cage, a minimal -regular graph of girth . In this construction, and are odd integers, in particular when is a power of 2 and we use the structure of generalized polygons. With this construction we obtain upper bounds for the -cages. Some of these graphs have the smallest number of vertices known so far among the regular graphs with girth . [Copyright &y& Elsevier]

Abstract: In this paper, we will present some of our recent results concerning the classical Erdős – Szekeres problem in combinatorial geometry [Copyright &y& Elsevier]

Abstract: A (k, g)-cage is a k-regular graph of girth g of minimum order. While many of the best known constructions of small k-regular graphs of girth g are known to be Cayley graphs, no general theory of the relation between the girth of a Cayley graph and the structure of the underlying group has been developed. We attempt to fill the gap by focusing on the girths of Caley graphs of nilpotent and solvable groups, and present a series of results supporting the intuitive idea that the closer a group is to being abelian, the less suitable it is for constructing Cayley graphs of large girths. Specifically, we establish the existence of upper bounds on the girths of Cayley graphs with respect to the nilpotency class and/or the length of the derived sequence of the underlying groups. [Copyright &y& Elsevier]

Abstract: De Bruijn and Erdös proved that every noncollinear set of n points in the plane determines at least n distinct lines. We suggest a possible generalization of this theorem in the framework of metric spaces and provide partial results on related extremal combinatorial problems. [Copyright &y& Elsevier]

Abstract: It was proved by Kahn [J. Kahn, Asymptotics of the chromatic index for multigraphs, Journal of Combinatorial Theory Series A 68 (1996) 233–254] that the chromatic number and fractional chromatic number of a line graph agree asymptotically. That is, for any line graph G we have . We extend this result to quasi-line graphs; G is a quasi-line graph if the neighbourhood of any vertex of G can be covered by two cliques. [Copyright &y& Elsevier]

Abstract: Given a graph , a positive integer k, and a positive integer d, we want find a subset with k vertices such the graph obtained by identifying the vertices from in G has diameter at most d. We prove that for every the problem is NP-complete. For the case of trees we provide a polynomial time algorithm that exploits the relationship with the r-dominating set problem. [Copyright &y& Elsevier]

Abstract: In this note, we present the main results of a series of forthcoming papers, dealing with bi-jective generalizations of some counting formulas. New intrinsic constructions in oriented matroids on a linearly ordered set of elements establish notably structural links between counting regions and linear programming. We introduce fully optimal bases, which have a simple combinatorial characterization, and strengthen the well-known optimal bases of linear programming. Our main result is that every bounded region of an ordered hyperplane arrangement, or ordered oriented matroid, has a unique fully optimal basis, providing the active bijection between bounded regions and uniactive internal bases. The active bijec-tion is extended to an activity preserving mapping between all reorientations and all bases of an ordered oriented matroid. It gives a bijective interpretation of the equality of two expressions for the Tutte polynomial, as well as a new expression of this polynomial in terms of beta invariants of minors. There are several refinements, such as an activity preserving bijection between regions (acyclic reorientations) and no-broken-circuit subsets, and others in terms of hyperplane arrangements, graphs, and permutations. [Copyright &y& Elsevier]

Abstract: The game chromatic number is considered for the Cartesian product of two graphs G and H. We determine exact values of when G and H belong to certain classes of graphs, and show that, in general, the game chromatic number is not bounded from above by a function of game chromatic numbers of graphs G and H. An analogous result is proved for the game coloring number of the Cartesian product. [Copyright &y& Elsevier]

Abstract: Let be the set of graphs with metric dimension β and diameter D. The first contribution is to characterize the graphs in with order for all values of β and D. The second contribution is to determine the maximum order of a graph in for all values of D and β. Only a weak upper bound was previously known. [Copyright &y& Elsevier]

Abstract: Let us consider a connected graph G with diameter D. For a given integer k between 0 and D, we say that G is k-walk-regular if the number of walks of length ℓ between vertices u, v only depends on the distance between u and v, provided that such a distance does not exceed k. Thus, in particular, a 0-walk-regular graph is the same as a walk-regular graph, where the number of cycles of length ℓ rooted at a given vertex is a constant through all the graph. In the other extreme, the distance-regular graphs correspond to the case . In this talk we discuss some algebraic characterizations of k-walk-regularity, in terms of the local spectrum and pre-distance-polynomials of G. Moreover, some results on the relationship between the diameter and the spectrum, as well as some geometric properties, of walk-regular graphs are presented. [Copyright &y& Elsevier]

Abstract: A k-cube (or “a unit cube in k dimensions”) is defined as the Cartesian product where (for ) is an interval of the form on the real line. The k-cube representation of a graph G is a mapping of the vertices of G to k-cubes such that the k-cubes mapped to two vertices in G have a non-empty intersection if and only if the vertices are adjacent. The cubicity of a graph G, denoted as cub(G), is defined as the minimum dimension k such that G has a k-cube representation. An interval graph is a graph that can be represented as the intersection of intervals on the real line - i.e., the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. We show that for any interval graph G with maximum degree Δ, . This upper bound is shown to be tight up to an additive constant of 4 by demonstrating interval graphs for which cubicity is equal to . [Copyright &y& Elsevier]

Abstract: We prove that every connected triangle-free graph on n vertices contains an induced tree on vertices, where c is a positive constant. The best known upper bound is . This partially answers questions of Erdős, Saks, and Sós and of Pultr. [Copyright &y& Elsevier]

Abstract: Let denote the minimal cardinality of a binary code of length n and covering radius one. Fundamental for the theory of lower bounds for is the covering excess method introduced by Johnson and v. Wee. Let δ i denote the covering excess on a sphere of radius i, . Generalizing an earlier result of v. Wee Habsieger and Honkala showed whenever for an odd prime p and holds. The author presents a new technique for proving the estimation instead. This answers a question of Habsieger and yields a “general improvement of the general excess bound” for binary codes with covering radius one. The proof uses a classification theorem for certain subset systems as well as new congruence properties for the δ-function, which were partially already conjectured by Habsieger. [Copyright &y& Elsevier]

Abstract: A simple topological graph is a drawing of a graph in the plane, where every two edges have at most one common point (an end-point or a crossing) and no three edges pass through a single crossing. Topological graphs G and H are isomorphic if H can be obtained from G by a homeomorphism of the sphere, and weakly isomorphic if G and H have the same set of pairs of crossing edges. We prove that the number of isomorphism classes of simple complete topological graphs on n vertices is . We also show that the number weak isomorphism classes of simple complete topological graphs with n vertices and crossings is at least , which improves the estimate of Harborth an Mengersen. [Copyright &y& Elsevier]

Abstract: We introduce a bijection between plane bipolar orientations with fixed numbers of vertices and faces, and non-intersecting triples of upright lattice paths with some specific extremities. Writing for the number of plane bipolar orientations with vertices and faces, our bijection provides a combinatorial proof of the following formula due to Baxter: [Copyright &y& Elsevier]

Abstract: For a non-hamiltonian claw-free graph G with order n and minimum degree δ we show the following. If , then G has a 2-factor with at most components, unless G belongs to a finite class of exceptional graphs. If , then G has a 2-factor with at most components. These bounds are sharp in the sense that we can replace nor 5/18 by a smaller quotient nor by δ. [Copyright &y& Elsevier]

Abstract: The Loebl–Komlós–Sós conjecture states that, given a graph G and a natural number k, if at least half the vertices of G have degree at least k, then any tree with at most k edges is a subgraph of G. We prove an approximate version of this conjecture for large graphs and k linear in . [Copyright &y& Elsevier]

Abstract: Fibonacci (alias Chebyshev) polynomials enjoy particular composition properties. These can be seen (and proved) from a combinatorial perspective by interpreting these polynomials as matching polynomials. An enumerative technique for cyclic structures is applied to obtain a generating polynomial identity for cyclic products of binomial coefficients in terms of matching polynomials. [Copyright &y& Elsevier]

Abstract: The packing chromatic number of a graph G is the smallest integer k such that the vertex set of G can be partitioned into k packings with pairwise different widths. Several lower and upper bounds are obtained for the packing chromatic number of Cartesian products of graphs. It is proved that the packing chromatic number of the infinite hexagonal lattice lies between 6 and 8. Optimal lower and upper bounds are proved for subdivision graphs. Trees are also considered and monotone colorings are introduced. [Copyright &y& Elsevier]

Abstract: We give a new and bijective proof for the formula of the growth function of the positive braid monoid with respect to Artin generators. [Copyright &y& Elsevier]

Abstract: For the unbiased Maker-Breaker game, played on the hypergraph , let be the smallest integer t such that Maker can win the game within t moves (if the game is a Breaker''s win, then set ). Similarly, for the unbiased Avoider-Enforcer game played on , let be the smallest integer t such that Enforcer can win the game within t moves (if the game is an Avoider''s win, then set ). We investigate and and determine their value for various positional games. [Copyright &y& Elsevier]

Abstract: An s-graph is a graph with two kind of edges: subdivisible edges and real edges. A realisation of an s-graph B is any graph obtained by subdividing subdivisible edges of B into paths of length at least one. Given an s-graph B, we study the decision problem . Its instance is any graph G, its question is “Does G contains a realisation of B as an induced subgraph ?”. For several B''s, the complexity is known and here we give the complexity for several more. We also provide results on the problem of detecting an induced cycle through two prescribed vertices. Display Omitted Display Omitted [Copyright &y& Elsevier]

Abstract: In this paper we present three different results dealing with the number of (⩽k)-facets of a set of points: [(i)] We give structural properties of sets in the plane that achieve the optimal lower bound of (⩽k)-edges for a fixed ; [(ii)] We show that the new lower bound for the number of (⩽k)-edges of a planar point set shown in [O. Aichholzer, J. García, D. Orden, and P. A. Ramos. New lower bounds for the number of (⩽k)-edges and the rectilinear crossing number of K. Discrete and Computational Geometry, in press] is optimal in the range ; [(iii)] We show that for the number of (⩽k)-facets of a set of n points in in general position is at least , and that this bound is tight in that range. [Copyright &y& Elsevier]

Abstract: Let be a positive integer and let denote the maximum number of edges in a -free simple graph of order ν. This paper gives the exact value of this function for all ν up to . This result allows us to deduce all the different values of the girths that such extremal graphs can have. Let be an integer. For each there exists ν such that every extremal graph G with has minimal degree at most 2, and is obtained by adding vertices of degree 1 and/or by subdividing a graph or a multigraph H with and . [Copyright &y& Elsevier]

Abstract: In this paper, we will give a short survey of various old and new results on Ramsey type problems in combinatorial geometry. [Copyright &y& Elsevier]

Abstract: In this paper, we will give a short survey of various old and new results concerning two closely connected problems of combinatorial geometry - that of K. Borsuk and that of E. Nelson - P. Erdös - H. Hadwiger. We will also present here our very recent achievements on Ramsey numbers. [Copyright &y& Elsevier]

Abstract: We consider combinatorial and computational issues that are related to the problem of moving coins from one configuration to another. Coins are defined as non-overlapping discs, and moves are defined as collision free translations, all in the Euclidean plane. We obtain combinatorial bounds on the number of moves that are necessary and/or sufficient to move coins from one configuration to another. We also consider several decision problems related to coin moving, and obtain some results regarding their computational complexity. [Copyright &y& Elsevier]

A long-standing open problem in combinatorial geometry is the chromatic number of the unit-distance graph in Rn; here points are adjacent if their distance in the ℓ2 norm is 1. For n=2, we know the answer is between 4 and 7. Little is known about other dimensions. The subgraphs induced by the rational points have been studied with limited success in small dimensions.We consider the analogous problem on the n-dimensional integer grid with fixed distance in the ℓ1 norm. That is, we make two integer grid points adjacent if the sum of the absolute differences in their coordinate values is r. Let the chromatic number of this graph be χ(Z,r).The main results of this paper are (i) χ(Zn,2)=2n for all n, and (ii) (1.139)n⩽χ(Zn,r)⩽(1/√ of 2πn)(5e)n for all n and even r. We also give bounds useful for small values of n and r. We also consider the lower and upper bounds on the n-dimensional real space with unit distance under ℓp norm for 1⩽p⩽∞. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain new upper estimates concerning the Erdős–Szekeres problem on empty hexagons. [Copyright &y& Elsevier]