Abstract: Publication date: June 2017 Source:European Journal of Combinatorics, Volume 63 Author(s): Catherine Greenhill, Mikhail Isaev, Matthew Kwan, Brendan D. McKay We give an asymptotic expression for the expected number of spanning trees in a random graph with a given degree sequence d = ( d 1 , … , d n ) , provided that the number of edges is at least n + 1 2 d max 4 , where d max is the maximum degree. A key part of our argument involves establishing a concentration result for a certain family of functions over random trees with given degrees, using Prüfer codes.

Abstract: Publication date: June 2017 Source:European Journal of Combinatorics, Volume 63 Author(s): Dan Hefetz, Michael Krivelevich, Wei En Tan We study two types of two player, perfect information games with no chance moves, played on the edge set of the binomial random graph G ( n , p ) . In each round of the ( 1 : q ) Waiter–Client Hamiltonicity game, the first player, called Waiter, offers the second player, called Client, q + 1 edges of G ( n , p ) which have not been offered previously. Client then chooses one of these edges, which he claims, and the remaining q edges go back to Waiter. Waiter wins this game if by the time every edge of G ( n , p ) has been claimed by some player, the graph consisting of Client’s edges is Hamiltonian; otherwise Client is the winner. Client–Waiter games are defined analogously, the main difference being that Client wins the game if his graph is Hamiltonian and Waiter wins otherwise. In this paper we determine a sharp threshold for both games. Namely, for every fixed positive integer q , we prove that the smallest edge probability p for which a.a.s. Waiter has a winning strategy for the ( 1 : q ) Waiter–Client Hamiltonicity game is ( 1 + o ( 1 ) ) log n / n , and the smallest p for which a.a.s. Client has a winning strategy for the ( 1 : q ) Client–Waiter Hamiltonicity game is ( q + 1 + o ( 1 ) ) log n / n .

Abstract: Publication date: Available online 3 March 2017 Source:European Journal of Combinatorics Author(s): Patrice Ossona de Mendez, Pierre Rosenstiehl

Abstract: Publication date: June 2017 Source:European Journal of Combinatorics, Volume 63 Author(s): Dongryul Kim The Golomb–Welch conjecture states that there are no perfect e -error-correcting codes in Z n for n ≥ 3 and e ≥ 2 . In this note, we prove the nonexistence of perfect 2 -error-correcting codes for a certain class of n , which is expected to be infinite. This result further substantiates the Golomb–Welch conjecture.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Shaofei Du, Xinyuan Hu A hypermap is a cellular embedding of a connected bipartite graph G into a compact and connected surface S without border. The vertices of G lie in distinct partitions, colored black and white, and are called respectively the hypervertices and hyperedges of the hypermap, while the connected regions of G ∖ S are the hyperfaces. A hypermap H is called orientable if the underlying surface S is orientable. An orientable hypermap is called regular if its orientation preserving automorphism group G acts regularly on the flags (hypervertex, hyperedge and hyperface incident triples), and further, it is called (face-)primer if G induces a faithful action on their hyperfaces. In Breda d’Azevedo and Fernandes (2011), a classification of the primer hypermap with a prime number of hyperfaces is given. In this paper, a classification will be given, for all primer hypermaps with a product of two primes hyperfaces, see Theorem 5.1.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Peter Frankl, Andrey Kupavskii Two families A and B of k -subsets of an n -set are called cross-intersecting if A ∩ B ≠ 0̸ for all A ∈ A , B ∈ B . Strengthening the classical Erdős–Ko–Rado theorem, Pyber proved that A B ≤ n − 1 k − 1 2 holds for n ≥ 2 k . In the present paper we sharpen this inequality. We prove that assuming B ≥ n − 1 k − 1 + n − i k − i + 1 for some 3 ≤ i ≤ k + 1 the stronger inequality A B ≤ ( n − 1 k − 1 + n − i k − i + 1 ) ( n − 1 k − 1 − n − i k − 1 ) holds. These inequalities are best possible. We also present a new short proof of Pyber’s inequality and a short computation-free proof of an inequality due to Frankl and Tokushige (1992).

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Nathan Williams We use a projection argument to uniformly prove that W -permutahedra and W -associahedra have the property that if v , v ′ are two vertices on the same face f , then any geodesic between v and v ′ does not leave f . In type A , we show that our geometric projection recovers a slight modification of the combinatorial projection in Sleator et al. (1988).

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Tara Fife, James Oxley A laminar family is a collection A of subsets of a set E such that, for any two intersecting sets, one is contained in the other. For a capacity function c on A , let I be { I : I ∩ A ≤ c ( A ) for all A ∈ A } . Then I is the collection of independent sets of a (laminar) matroid on E . We present a method of compacting laminar presentations, characterize the class of laminar matroids by their excluded minors, present a way to construct all laminar matroids using basic operations, and compare the class of laminar matroids to other well-known classes of matroids.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Zuzana Masáková, Edita Pelantová, Štěpán Starosta We study purely morphic words coding symmetric non-degenerate three interval exchange transformation which are known to be palindromic, i.e., they contain infinitely many palindromes. We prove that such words are fixed by a conjugate to a morphism of class P , that is, a morphism such that each letter a is mapped to p p a where p and p a are both palindromes. We thus provide a new family of palindromic infinite words satisfying the conjecture of Hof, Knill and Simon. Given a morphism fixing such word, we give a formula to determine the parameters of the underlying three interval exchange and the intercept of the word.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Sebastian M. Cioabă, Jack Koolen, Weiqiang Li A connected graph G of even order v is called t -extendable if it contains a perfect matching, t < v / 2 and any matching of t edges is contained in some perfect matching. The extendability of G is the maximum t such that G is t -extendable. Since its introduction by Plummer in the 1980s, this combinatorial parameter has been studied for many classes of interesting graphs. In 2005, Brouwer and Haemers proved that every distance-regular graph of even order is 1 -extendable and in 2014, Cioabă and Li showed that any connected strongly regular graph of even order is 3-extendable except for a small number of exceptions. In this paper, we extend and generalize these results. We prove that all distance-regular graphs with diameter D ≥ 3 are 2-extendable and we also obtain several better lower bounds for the extendability of distance-regular graphs of valency k ≥ 3 that depend on k , λ and μ , where λ is the number of common neighbors of any two adjacent vertices and μ is the number of common neighbors of any two vertices in distance two. In many situations, we show that the extendability of a distance-regular graph with valency k grows linearly in k . We conjecture that the extendability of a distance-regular graph of even order and valency k is at least ⌈ k / 2 ⌉ − 1 and we prove this fact for bipartite distance-regular graphs. In course of this investigation, we obtain some new bounds for the max-cut and the independence number of distance-regular graphs in terms of their size and odd girth and we prove that our inequalities are incomparable with known eigenvalue bounds for these combinatorial parameters.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Chris Godsil, Brendan Rooney Computing the clique number and chromatic number of a general graph are well-known NP-Hard problems. Codenotti et al., (1998) showed that computing clique number and chromatic number are both NP-Hard problems for the class of circulant graphs. We show that computing clique number is NP-Hard for the class of Cayley graphs for the groups G n , where G is any fixed finite group (e.g., cubelike graphs). We also show that computing chromatic number cannot be done in polynomial time (under the assumption NP ≠ ZPP ) for the same class of graphs. Our presentation uses free Cayley graphs. The proof combines free Cayley graphs with quotient graphs and Goppa codes.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Evgeniy Krasko, Alexander Omelchenko We obtain explicit formulas for enumerating rooted and unrooted 4 -regular one-face maps on genus g surfaces. For rooted maps the result is combinatorially derived from Chapuy’s vertex cutting bijection and has a simple sum-free form similar to analogous formulas for general and cubic one-face maps. To enumerate unrooted maps we apply the approach of Liskovets, Mednykh and Nedela of reducing the problem to counting rooted maps on orbifolds. We show that for 4 -regular one-face maps the set of orbifolds to be considered has a simple description, which allows us to obtain the final formula in an explicit form of a sum over 3 integer indices.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Richard P. Stanley Let JT λ be the Jacobi–Trudi matrix corresponding to the partition λ , so det JT λ is the Schur function s λ in the variables x 1 , x 2 , … . Set x 1 = ⋯ = x n = 1 and all other x i = 0 . Then the entries of JT λ become polynomials in n of the form ( n + j − 1 j ) . We determine the Smith normal form over the ring Q [ n ] of this specialization of JT λ . The proof carries over to the specialization x i = q i − 1 for 1 ≤ i ≤ n and x i = 0 for i > n , where we set q n = y and work over the ring Q ( q ) [ y ] .

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Ilan Karpas, Eoin Long A balanced pattern of order 2 d is an element P ∈ { + , − } 2 d , where both signs appear d times. Two sets A , B ⊂ [ n ] form a P -pattern, which we denote by pat ( A , B ) = P , if A △ B = { j 1 , … , j 2 d } with 1 ≤ j 1 < ⋯ < j 2 d ≤ n and { i ∈ [ 2 d ] : P i = + } = { i ∈ [ 2 d ] : j i ∈ A ∖ B } . We say A ⊂ P [ n ] is P -free if pat ( A , B ) ≠ P for all A , B ∈ A . We consider the following extremal question: how large can a family A ⊂ P [ n ] be if A is P -free? We prove a number of results on the sizes of such families. In particular, we show that for some fixed c > 0 , if P is a d -balanced pattern with d < c log log n then ∣ A ∣ = o ( 2 n ) . We then give stronger bounds in the cases when (i) P consists of d + signs, followed by d − signs and (ii) P consists of alternating signs. In both cases, if d = o ( n ) then ∣ A ∣ = o ( 2 n ) . In the case of (i), this is tight.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Dhruv Mubayi Given integers ℓ , n , the ℓ th power of the path P n is the ordered graph P n ℓ with vertex set v 1 < v 2 < ⋯ < v n and all edges of the form v i v j where i − j ≤ ℓ . The Ramsey number r ( P n ℓ , P n ℓ ) is the minimum N such that every 2-coloring of [ N ] 2 results in a monochromatic copy of P n ℓ . It is well-known that r ( P n 1 , P n 1 ) = ( n − 1 ) 2 + 1 . For ℓ > 1 , Balko–Cibulka–Král–Kynčl proved that r ( P n ℓ , P n ℓ ) < c ℓ n 128 ℓ and asked for the growth rate for fixed ℓ . When ℓ = 2 , we improve this upper bound substantially by proving r ( P n 2 , P n 2 ) < c n 19.5 . Using this result, we determine the correct tower growth rate of the k -uniform hypergraph Ramsey number of a ( k + 1 ) -clique versus an ordered tight path. Finally, we consider an ordered version of the classical Erdős–Hajnal hypergraph Ramsey problem, improve the tower height given by the trivial upper bound, and conjecture that this tower height is optimal.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Anurag Bishnoi, Bart De Bruyn We prove that there are no semi-finite generalized hexagons with q + 1 points on each line containing the known generalized hexagons of order q as full subgeometries when q is equal to 3 or 4 , thus contributing to the existence problem of semi-finite generalized polygons posed by Tits. The case when q is equal to 2 was treated by us in an earlier work, for which we give an alternate proof. For the split Cayley hexagon of order 4 we obtain the stronger result that it cannot be contained as a proper full subgeometry in any generalized hexagon.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): József Solymosi, Ching Wong Consider a family of graphs having a fixed girth and a large size. We give an optimal lower asymptotic bound on the number of even cycles of any constant length, as the order of the graphs tends to infinity.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Sébastien Labbé, Edita Pelantová, Štěpán Starosta Brlek et al., conjectured in 2008 that any fixed point of a primitive morphism with finite palindromic defect is either periodic or its palindromic defect is zero. Bucci and Vaslet disproved this conjecture in 2012 by a counterexample over ternary alphabet. We prove that the conjecture is valid on binary alphabet. We also describe a class of morphisms over multiliteral alphabet for which the conjecture still holds. The proof is based on properties of extension graphs.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Emma Cohen, Péter Csikvári, Will Perkins, Prasad Tetali Let H WR be the path on 3 vertices with a loop at each vertex. D. Galvin (Galvin, 2013, 2014) conjectured, and E. Cohen, W. Perkins and P. Tetali (Cohen et al., in press) proved that for any d -regular simple graph G on n vertices we have hom ( G , H WR ) ≤ hom ( K d + 1 , H WR ) n / ( d + 1 ) . In this paper we give a short proof of this theorem together with the proof of a conjecture of Cohen, Perkins and Tetali (Cohen et al., in press). Our main tool is a simple bijection between the Widom–Rowlinson model and the hard-core model on another graph. We also give a large class of graphs H for which we have hom ( G , H ) ≤ hom ( K d + 1 , H ) n / ( d + 1 ) . In particular, we show that the above inequality holds if H is a path or a cycle of even length at least 6 with loops at every vertex.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Nicola Durante, Rocco Trombetti, Yue Zhou In each of the three projective planes coordinatized by the Knuth’s binary semifield K n of order 2 n and two of its Knuth derivatives, we exhibit a new infinite family of translation hyperovals. In particular, when n = 5 , we also present complete lists of all translation hyperovals in them. The properties of some designs associated with these hyperovals are also studied.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Andrei Asinowski, Christian Krattenthaler, Toufik Mansour We compute the number of triangulations of a convex k -gon each of whose sides is subdivided by r − 1 points. We find explicit formulas and generating functions, and we determine the asymptotic behavior of these numbers as k and/or r tend to infinity. We connect these results with the question of finding the planar set of points in general position that has the minimum possible number of triangulations — a well-known open problem from computational geometry.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Anne Schilling, Nicolas M. Thiéry, Graham White, Nathan Williams We prove that the expected number of braid moves in the commutation class of the reduced word ( s 1 s 2 ⋯ s n − 1 ) ( s 1 s 2 ⋯ s n − 2 ) ⋯ ( s 1 s 2 ) ( s 1 ) for the long element in the symmetric group S n is one. This is a variant of a similar result by V. Reiner, who proved that the expected number of braid moves in a random reduced word for the long element is one. The proof is bijective and uses X. Viennot’s theory of heaps and variants of the promotion operator. In addition, we provide a refinement of this result on orbits under the action of even and odd promotion operators. This gives an example of a homomesy for a nonabelian (dihedral) group that is not induced by an abelian subgroup. Our techniques extend to more general posets and to other statistics.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Primož Potočnik, Steve Wilson A new product construction of graphs and digraphs, called the separated box product, is presented, and several of its properties are discussed. The construction is based on the standard box product of digraphs. However, unlike the standard box product, it preserves the valence of the factor digraphs: If every vertex in both factors has in-valence and out-valence k for some fixed k , then so does every vertex of the separated box product. Questions about the symmetries of the product and their relations to symmetries of the factor graphs are considered. An application of this construction to the case of tetravalent edge-transitive graphs is discussed in detail.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Pablo Torres, Mario Valencia-Pabon A graph G is said to be hom-idempotent if there is a homomorphism from G 2 to G , and weakly hom-idempotent if for some n ≥ 1 there is a homomorphism from G n + 1 to G n . Larose et al. (1998) proved that Kneser graphs KG ( n , k ) are not weakly hom-idempotent for n ≥ 2 k + 1 , k ≥ 2 . For s ≥ 2 , we characterize all the shifts (i.e., automorphisms of the graph that map every vertex to one of its neighbors) of s -stable Kneser graphs KG ( n , k ) s − stab and we show that 2 -stable Kneser graphs are not weakly hom-idempotent, for n ≥ 2 k + 2 , k ≥ 2 . Moreover, for s , k ≥ 2 , we prove that s -stable Kneser graphs KG ( k s + 1 , k ) s − stab are circulant graphs and so hom-idempotent graphs. Finally, for s ≥ 3 , we show that s -stable Kneser graphs KG ( 2 s + 2 , 2 ) s − stab are cores, not χ -critical, not hom-idempotent and their chromatic number is equal to s + 2 .

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): B. Miraftab, M.J. Moghadamzadeh A problem by Diestel is to extend algebraic flow theory of finite graphs to infinite graphs with ends. In order to pursue this problem, we define an A -flow and non-elusive H -flow for arbitrary graphs and for abelian Hausdorff topological groups H and compact subsets A ⊆ H . We use these new definitions to extend several well-known theorems of flows in finite graphs to infinite graphs.

Abstract: Publication date: May 2017 Source:European Journal of Combinatorics, Volume 62 Author(s): Louis Esperet, Laetitia Lemoine, Frédéric Maffray We prove that every 3-connected planar graph on n vertices contains an induced path on Ω ( log n ) vertices, which is best possible and improves the best known lower bound by a multiplicative factor of log log n . We deduce that any planar graph (or more generally, any graph embeddable on a fixed surface) with a path on n vertices, also contains an induced path on Ω ( log n ) vertices. We conjecture that for any k , there is a positive constant c ( k ) such that any k -degenerate graph with a path on n vertices also contains an induced path on Ω ( ( log n ) c ( k ) ) vertices. We provide examples showing that this order of magnitude would be best possible (already for chordal graphs), and prove the conjecture in the case of interval graphs.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Payam Valadkhan Given a symmetric m × m matrix M with entries from the set { 0 , 1 , ∗ } , the M -edge-partition problem asks whether the edges of a given graph can be partitioned into m parts E 0 , E 1 ⋯ E m − 1 such that any two distinct edges in (possibly equal) parts V i and V j have a common endpoint if M ( i , j ) = 1 , and no common endpoint if M ( i , j ) = 0 . This problem generalizes some well-known edge-partition problems (such as the edge-coloring problem), and is in close relation with the well-known M -partition problem introduced by Feder et al. Following the current trends, we prove some complexity results for the list version of the M -edge-partition problem restricted to simple graphs.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Pablo Spiga, Gabriel Verret We consider vertex-primitive digraphs having two vertices with almost equal neighbourhoods (that is, the set of vertices that are neighbours of one but not the other is small). We prove a structural result about such digraphs and then apply it to answer a question of Araújo and Cameron about synchronising groups.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Alin Bostan, Frédéric Chyzak, Mark van Hoeij, Manuel Kauers, Lucien Pech We study nearest-neighbors walks on the two-dimensional square lattice, that is, models of walks on Z 2 defined by a fixed step set that is a subset of the non-zero vectors with coordinates 0, 1 or − 1 . We concern ourselves with the enumeration of such walks starting at the origin and constrained to remain in the quarter plane N 2 , counted by their length and by the position of their ending point. Bousquet-Mélou and Mishna (2010) identified 19 models of walks that possess a D-finite generating function; linear differential equations have then been guessed in these cases by Bostan and Kauers (2009). We give here the first proof that these equations are indeed satisfied by the corresponding generating functions. As a first corollary, we prove that all these 19 generating functions can be expressed in terms of Gauss’ hypergeometric functions that are intimately related to elliptic integrals. As a second corollary, we show that all the 19 generating functions are transcendental, and that among their 19 × 4 combinatorially meaningful specializations only four are algebraic functions.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Dmitry N. Kozlov The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on the round complexity of the weak symmetry breaking task. Furthermore, we introduce the notion of a fundamental binomial identity and find an infinite family of values, other than the prime powers, for which no fundamental binomial identity can exist.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Nickolas Hein, Jia Huang The Catalan number C n enumerates parenthesizations of x 0 ∗ ⋯ ∗ x n where ∗ is a binary operation. We introduce the modular Catalan number C k , n to count equivalence classes of parenthesizations of x 0 ∗ ⋯ ∗ x n when ∗ satisfies a k -associative law generalizing the usual associativity. This leads to a study of restricted families of Catalan objects enumerated by C k , n with emphasis on binary trees, plane trees, and Dyck paths, each avoiding certain patterns. We give closed formulas for C k , n with two different proofs. For each n ≥ 0 we compute the largest size of k -associative equivalence classes and show that the number of classes with this size is a Catalan number.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Fabrizio Caselli, Mario Marietti Special matchings are purely combinatorial objects associated with a partially ordered set, which have applications in Coxeter group theory. We provide an explicit characterization and a complete classification of all special matchings of any lower Bruhat interval. The results hold in any arbitrary Coxeter group and have also applications in the study of the corresponding parabolic Kazhdan–Lusztig polynomials.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Matt DeVos, Daryl Funk, Irene Pivotto We investigate the set of excluded minors of connectivity 2 for the class of frame matroids. We exhibit a list E of 18 such matroids, and show that if N is such an excluded minor, then either N ∈ E or N is a 2-sum of U 2 , 4 and a 3-connected non-binary frame matroid.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Tom Kelly, Chun-Hung Liu A feedback vertex set of a graph is a subset of vertices intersecting all cycles. We provide tight upper bounds on the size of a minimum feedback vertex set in planar graphs of girth at least five. We prove that if G is a connected planar graph of girth at least five on n vertices and m edges, then G has a feedback vertex set of size at most 2 m − n + 2 7 . By Euler’s formula, this implies that G has a feedback vertex set of size at most m 5 and n − 2 3 . These results not only improve a result of Dross, Montassier and Pinlou and confirm the girth-5 case of one of their conjectures, but also make the best known progress towards a conjecture of Kowalik, Lužar and Škrekovski and solve the subcubic case of their conjecture. An important step of our proof is providing an upper bound on the size of minimum feedback vertex sets of subcubic graphs with girth at least five with no induced subdivision of members of a finite family of non-planar graphs.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Nevena Francetić, Sarada Herke, Brendan D. McKay, Ian M. Wanless Ryser conjectured that τ ⩽ ( r − 1 ) ν for r -partite hypergraphs, where τ is the covering number and ν is the matching number. We prove this conjecture for r ⩽ 9 in the special case of linear intersecting hypergraphs, in other words where every pair of lines meets in exactly one vertex. Aharoni formulated a stronger version of Ryser’s conjecture which specified that each r -partite hypergraph should have a cover of size ( r − 1 ) ν of a particular form. We provide a counterexample to Aharoni’s conjecture with r = 13 and ν = 1 . We also report a number of computational results. For r = 7 , we find that there is no linear intersecting hypergraph that achieves the equality τ = r − 1 in Ryser’s conjecture, although non-linear examples are known. We exhibit intersecting non-linear examples achieving equality for r ∈ { 9 , 13 , 17 } . Also, we find that r = 8 is the smallest value of r for which there exists a linear intersecting r -partite hypergraph that achieves τ = r − 1 and is not isomorphic to a subhypergraph of a projective plane.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Mathias Pétréolle An element of a Coxeter group is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied by Boothby et al. (2012). In particular the authors precisely identified the Coxeter groups having a finite number of cyclically fully commutative elements and enumerated them. In this work we characterize and enumerate those elements according to their Coxeter length in all finite and all affine Coxeter groups by using an operation on heaps, the cylindrical closure. In finite types, this refines the work of Boothby et al. (2012), by adding a new parameter. In affine type, all the results are new. In particular, we prove that there is a finite number of cyclically fully commutative logarithmic elements in all affine Coxeter groups. We also study the cyclically fully commutative involutions and prove that their number is finite in all Coxeter groups.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Bhaskar Bagchi, Basudeb Datta, Jonathan Spreer For a field F , the notion of F -tightness of simplicial complexes was introduced by Kühnel. Kühnel and Lutz conjectured that F -tight triangulations of a closed manifold are the most economic of all possible triangulations of the manifold. The boundary of a triangle is the only F -tight triangulation of a closed 1-manifold. A triangulation of a closed 2-manifold is F -tight if and only if it is F -orientable and neighbourly. In this paper we prove that a triangulation of a closed 3-manifold is F -tight if and only if it is F -orientable, neighbourly and stacked. In consequence, the Kühnel–Lutz conjecture is valid in dimension ≤ 3 .

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Hiroshi Maehara, Horst Martini Let X be an infinite set in R d that has no accumulation point. We prove that the following statement holds for each d -dimensional polyhedron Π , i.e., for each bounded part of R d generated by a closed polyhedral surface: for any positive integer n , there is a polyhedron similar to Π that contains exactly n points taken from X .

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Hiroshi Maehara, Horst Martini Let X be an infinite set in R d that has no accumulation point. We prove that the following statement holds for each d -dimensional polyhedron Π , i.e., for each bounded part of R d generated by a closed polyhedral surface: for any positive integer n , there is a polyhedron similar to Π that contains exactly n points taken from X .

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Jesper M. Møller The first part of this paper deals with the combinatorics of equivariant partitions of finite sets with group actions. In the second part, we compute all equivariant Euler characteristics of the Σ n -poset of non-extreme partitions of an n -set.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Darren B. Glass In this paper we consider the critical group of finite connected graphs which admit harmonic actions by the dihedral group D n , extending earlier work by the author and Criel Merino. In particular, we show that the critical group of such a graph can be decomposed in terms of the critical groups of the quotients of the graph by certain subgroups of the automorphism group. This is analogous to a theorem of Kani and Rosen which decomposes the Jacobians of algebraic curves with a D n -action.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Huda Chuangpishit, Mahya Ghandehari, Jeannette Janssen Let w : [ 0 , 1 ] 2 → [ 0 , 1 ] be a symmetric function, and consider the random process G ( n , w ) , where vertices are chosen from [ 0 , 1 ] uniformly at random, and w governs the edge formation probability. Such a random graph is said to have a linear embedding, if the probability of linking to a particular vertex v decreases with distance. The rate of decrease, in general, depends on the particular vertex v . A linear embedding is called uniform if the probability of a link between two vertices depends only on the distance between them. In this article, we consider the question whether it is possible to “transform” a linear embedding to a uniform one, through replacing the uniform probability space [ 0 , 1 ] with a suitable probability space on R . We give necessary and sufficient conditions for the existence of a uniform linear embedding for random graphs where w attains only a finite number of values. Our findings show that for a general w the answer is negative in most cases.

Abstract: Publication date: March 2017 Source:European Journal of Combinatorics, Volume 61 Author(s): Wenjie Fang, Louis-François Préville-Ratelle Let v be a grid path made of north and east steps. The lattice Tam(v), based on all grid paths weakly above v and sharing the same endpoints as v , was introduced by Préville-Ratelle and Viennot (2016) and corresponds to the usual Tamari lattice in the case v = ( N E ) n . Our main contribution is that the enumeration of intervals in Tam(v), over all v of length n , is given by 2 ( 3 n + 3 ) ! ( n + 2 ) ! ( 2 n + 3 ) ! . This formula was first obtained by Tutte (1963) for the enumeration of non-separable planar maps. Moreover, we give an explicit bijection from these intervals in Tam(v) to non-separable planar maps.

Abstract: Publication date: February 2017 Source:European Journal of Combinatorics, Volume 60 Author(s): Raphael Yuster Given positive integers h and k , denote by r ( h , k ) the smallest integer n such that in any k -coloring of the edges of a tournament on more than n vertices there is a monochromatic copy of every oriented tree on h vertices. We prove that r ( h , k ) = ( h − 1 ) k for all k sufficiently large ( k = Θ ( h log h ) suffices). The bound ( h − 1 ) k is tight. The related parameter r ∗ ( h , k ) where some color contains all oriented trees is asymptotically determined. Values of r ( h , 2 ) for some small h are also established.

Abstract: Publication date: February 2017 Source:European Journal of Combinatorics, Volume 60 Author(s): Carolyn Chun, Rhiannon Hall, Criel Merino, Steven Noble We develop some basic tools to work with representable matroids of bounded tree-width and use them to prove that, for any prime power q and constant k , the characteristic polynomial of any loopless, G F ( q ) -representable matroid with tree-width k has no real zero greater than q k − 1 .

Abstract: Publication date: February 2017 Source:European Journal of Combinatorics, Volume 60 Author(s): Sławomir Solecki, Min Zhao We prove a Ramsey theorem for finite sets equipped with a partial order and a fixed number of linear orders extending the partial order. This is a common generalization of two recent Ramsey theorems due to Sokić. As a bonus, our proof gives new arguments for these two results.

Abstract: Publication date: February 2017 Source:European Journal of Combinatorics, Volume 60 Author(s): Shoni Gilboa, Dan Hefetz The degree anti-Ramsey number A R d ( H ) of a graph H is the smallest integer k for which there exists a graph G with maximum degree at most k such that any proper edge colouring of G yields a rainbow copy of H . In this paper we prove a general upper bound on degree anti-Ramsey numbers, determine the precise value of the degree anti-Ramsey number of any forest, and prove an upper bound on the degree anti-Ramsey numbers of cycles of any length which is best possible up to a multiplicative factor of 2 . Our proofs involve a variety of tools, including a classical result of Bollobás concerning cross intersecting families and a topological version of Hall’s Theorem due to Aharoni, Berger and Meshulam.

Abstract: Publication date: February 2017 Source:European Journal of Combinatorics, Volume 60 Author(s): Richard Lang, Maya Stein We show that for any 2-local colouring of the edges of the balanced complete bipartite graph K n , n , its vertices can be covered with at most 3 disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite r -locally coloured graph with O ( r 2 ) disjoint monochromatic cycles. We also determine the 2-local bipartite Ramsey number of a path almost exactly: Every 2-local colouring of the edges of K n , n contains a monochromatic path on n vertices.