Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Louis Esperet, Jean-Florent Raymond Let C be a class of graphs that is closed under taking subgraphs. We prove that if for some fixed 0 < δ ≤ 1 , every n -vertex graph of C has a balanced separator of order O ( n 1 − δ ) , then any depth- r minor (i.e. minor obtained by contracting disjoint subgraphs of radius at most r ) of a graph in C has average degree O ( ( r polylog r ) 1 ∕ δ ) . This confirms a conjecture of Dvořák and Norin.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Jakub Bulín For a fixed digraph H , the H -coloring problem is the problem of deciding whether a given input digraph G admits a homomorphism to H . The CSP dichotomy conjecture of Feder and Vardi is equivalent to proving that, for any H , the H -coloring problem is in P or NP-complete. We confirm this dichotomy for a certain class of oriented trees, which we call special trees (generalizing earlier results on special triads and polyads). Moreover, we prove that every tractable special oriented tree has bounded width, i.e., the corresponding H -coloring problem is solvable by local consistency checking. Our proof relies on recent algebraic tools, namely characterization of congruence meet-semidistributivity via pointing operations and absorption theory.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Yvonne Kemper, Jim Lawrence Questions of the existence of Hamiltonian circuits in the tope graphs of central arrangements of hyperplanes are considered. Connections between the existence of Hamiltonian circuits in the arrangement and the odd–even invariant of the arrangement are described. Some new results concerning bounds on the odd–even invariant are obtained. All results can be formulated more generally for oriented matroids and are still valid in that setting.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Vladimir Turaev We introduce and study additive posets. We show that the top homology group (with coefficients in Z ∕ 2 Z ) of a finite dimensional CW-complex carries a structure of an additive poset invariant under subdivisions. Applications to CW-complexes and graphs are discussed.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Thomas Blankenship, Jay Cummings, Vladislav Taranchuk In this paper we prove a new recurrence relation on the van der Waerden numbers, w ( r , k ) . In particular, if p is a prime and p ≤ k then w ( r , k ) > p ⋅ w r − r p , k − 1 . This recurrence gives the lower bound w ( r , p + 1 ) > p r − 1 2 p when r ≤ p , which generalizes Berlekamp’s theorem on 2-colorings, and gives the best known bound for a large interval of r . The recurrence can also be used to construct explicit valid colorings, and it improves known lower bounds on small van der Waerden numbers.

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): S. Artamonov, M. Babenko A perfect 2-matching in an undirected graph G = ( V , E ) is a function x : E → 0 , 1 , 2 such that for each node v ∈ V the sum of values x ( e ) on all edges e incident to v equals 2. If supp ( x ) = e ∈ E ∣ x ( e ) ≠ 0 contains no triangles then x is called triangle-free. Polyhedrally speaking, triangle-free 2-matchings are harder than 2-matchings, but easier than usual 1-matchings. Given edge costs c : E → R + , a natural combinatorial problem consists in finding a perfect triangle-free matching of minimum total cost. For this problem, Cornuéjols and Pulleyblank devised a combinatorial strongly-polynomial algorithm, which can be implemented to run in O ( V E log V ) time. (Here we write V , E to indicate their cardinalities V , E .) If edge costs are integers in range [ 0 , C ] then for both 1- and 2-matchings some faster scaling algorithms are known that find optimal solutions within O ( V α ( E , V ) log V E log ( V C ) ) and O ( V E log ( V C ) ) time, respectively, where α denotes the inverse Ackermann function. So far, no efficient cost-scaling algorithm is known for finding a minimum-cost perfect triangle-free2-matching. The present paper fills this gap by presenting such an algorithm with time complexity of O ( V E log V log ( V C ) ) .

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): Carla Binucci, Emilio Di Giacomo, Md. Iqbal Hossain, Giuseppe Liotta A drawing of a graph such that the vertices are drawn as points along a line and each edge is a circular arc in one of the two half-planes defined by this line is called a 2 -page drawing. If all edges are in the same half-plane, the drawing is called a 1 -page drawing. We want to compute 1 -page and 2 -page drawings of planar graphs such that the number of crossings per edge does not depend on the number of vertices. We show that for any constant k , there exist planar graphs that require more than k crossings per edge in both 1 -page and 2 -page drawings. We then prove that if the vertex degree is bounded by Δ , every planar 3-tree has a 2 -page drawing with a number of crossings per edge that only depends on Δ . Finally, we show a similar result for 1 -page drawings of partial 2 -trees.

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): Ilkyoo Choi, Jan Ekstein, Přemysl Holub, Bernard Lidický Given a triangle-free planar graph G and a 9 -cycle C in G , we characterize situations where a 3 -coloring of C does not extend to a proper 3 -coloring of G . This extends previous results when C is a cycle of length at most 8 .

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): Daniel Freund, Matthias Poloczek, Daniel Reichman We study the activation process in undirected graphs known as bootstrap percolation: a vertex is active either if it belongs to a set of initially activated vertices or if at some point it had at least r active neighbors, for a threshold r that is identical for all vertices. A contagious set is a vertex set whose activation results with the entire graph being active. Let m ( G , r ) be the size of a smallest contagious set in a graph G on n vertices. We examine density conditions that ensure m ( G , r ) = r for all r , and first show a necessary and sufficient condition on the minimum degree. Moreover, we study the speed with which the activation spreads and provide tight upper bounds on the number of rounds it takes until all nodes are activated in such graphs. We also investigate what average degree asserts the existence of small contagious sets. For n ≥ k ≥ r , we denote by M ( n , k , r ) the maximum number of edges in an n -vertex graph G satisfying m ( G , r ) > k . We determine the precise value of M ( n , k , 2 ) and M ( n , k , k ) , assuming that n is sufficiently large compared to k .

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): Nils Kriege, Florian Kurpicz, Petra Mutzel The complexity of the maximum common connected subgraph problem in partial k -trees is still not fully understood. Polynomial-time solutions are known for degree-bounded outerplanar graphs, a subclass of the partial 2 -trees. On the other hand, the problem is known to be NP-hard in vertex-labeled partial 11 -trees of bounded degree. We consider series–parallel graphs, i.e., partial 2 -trees. We show that the problem remains NP-hard in biconnected series–parallel graphs with all but one vertex of degree 3 or less. A positive complexity result is presented for a related problem of high practical relevance which asks for a maximum common connected subgraph that preserves blocks and bridges of the input graphs. We present a polynomial time algorithm for this problem in series–parallel graphs, which utilizes a combination of BC- and SP-tree data structures to decompose both graphs.

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): Fan Chung, Olivia Simpson Heat kernel pagerank is a variation of Personalized PageRank given in an exponential formulation. In this work, we present a sublinear time algorithm for approximating the heat kernel pagerank of a graph. The algorithm works by simulating random walks of bounded length and runs in time O ( log ( ϵ − 1 ) log n ϵ 3 log log ( ϵ − 1 ) ) , assuming performing a random walk step and sampling from a distribution with bounded support take constant time. The quantitative ranking of vertices obtained with heat kernel pagerank can be used for local clustering algorithms. We present an efficient local clustering algorithm that finds cuts by performing a sweep over a heat kernel pagerank vector, using the heat kernel pagerank approximation algorithm as a subroutine. Specifically, we show that for a subset S of Cheeger ratio ϕ , many vertices in S may serve as seeds for a heat kernel pagerank vector which will find a cut of conductance O ( ϕ ) .

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): Saad I. El-Zanati, Uthoomporn Jongthawonwuth, Heather Jordon, Charles Vanden Eynden Let G of order n be the vertex-disjoint union of two cycles. It is known that there exists a G -decomposition of K v for all v ≡ 1 ( mod 2 n ) . If G is bipartite and x is a positive integer, it is also known that there exists a G -decomposition of K n x − I , where I is a 1-factor. If G is not bipartite, there exists a G -decomposition of K n if n is odd, and of K n − I , where I is a 1-factor, if n is even. We use novel extensions of the Bose construction for Steiner triple systems and some recent results on the Oberwolfach Problem to obtain a G -decomposition of K v for all v ≡ n ( mod 2 n ) when n is odd, unless G = C 4 ∪ C 5 and v = 9 . If G consists of two odd cycles and n ≡ 0 ( mod 4 ) , we also obtain a G -decomposition of K v − I , for all v ≡ 0 ( mod n ) , v ≠ 4 n .

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): Petr A. Golovach, Pinar Heggernes, Dieter Kratsch Connected Vertex Cover is one of the classical problems of computer science, already mentioned in the monograph of Garey and Johnson (1979). Although the optimization and decision variants of finding connected vertex covers of minimum size or weight are well-studied, surprisingly there is no work on the enumeration or maximum number of minimal connected vertex covers of a graph. In this paper we show that the number of minimal connected vertex covers of a graph is at most 1 . 866 8 n , and these sets can be enumerated in time O ( 1 . 866 8 n ) . For graphs of chordality at most 5, we are able to give a better upper bound, and for chordal graphs and distance-hereditary graphs we are able to give tight bounds on the maximum number of minimal connected vertex covers.

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): Klaus Jansen, Stefan E.J. Kraft The Unbounded Knapsack Problem (UKP) is a well-known variant of the famous 0–1 Knapsack Problem (0–1 KP). In contrast to 0–1 KP, an arbitrary number of copies of every item can be taken in UKP. Since UKP is NP-hard, fully polynomial time approximation schemes (FPTAS) are of great interest. Such algorithms find a solution arbitrarily close to the optimum OPT ( I ) , i.e., of value at least ( 1 − ε ) OPT ( I ) for ε > 0 , and have a running time polynomial in the input length and 1 ε . For over thirty years, the best FPTAS was due to Lawler with running time O ( n + 1 ε 3 ) and space complexity O ( n + 1 ε 2 ) , where n is the number of knapsack items. We present an improved FPTAS with running time O ( n + 1 ε 2 log 3 1 ε ) and space bound O ( n + 1 ε log 2 1 ε ) . This directly improves the running time of the fastest known approximation schemes for Bin Packing and Strip Packing, which have to approximately solve UKP instances as subproblems.

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): Meirav Zehavi The klam value of an algorithm that runs in time O ∗ ( f ( k ) ) is the maximal value k such that f ( k ) < 1 0 20 . Given a graph G and a parameter k , the k -Leaf Spanning Tree ( k -LST) problem asks if G contains a spanning tree with at least k leaves. This problem has been extensively studied over the past three decades. In 2000, Fellows et al. (2000) asked whether it admits a klam value of 50. A steady progress towards an affirmative answer continued until 5 years ago, when an algorithm of klam value 37 was discovered. Our contribution is twofold. First, we present an O ∗ ( 3 . 18 8 k ) -time parameterized algorithm for k -LST, which shows that the problem admits a klam value of 39. Second, we rely on an application of the bounded search trees technique where the correctness of rules crucially depends on the history of previously applied rules in a non-standard manner, encapsulated in a “dependency claim”. Similar claims may be used to capture the essence of other complex algorithms in a compact, useful manner.

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): F. Blanchet-Sadri, M. Cordier, R. Kirsch We consider the border sets of partial words and study the combinatorics of specific representations of them, called border correlations, which are binary vectors of same length indicating the borders. We characterize precisely which of these vectors are valid border correlations, and establish a one-to-one correspondence between the set of valid border correlations and the set of valid ternary period correlations of a given length, the latter being ternary vectors representing the strong and strictly weak period sets. It turns out that the sets of all border correlations of a given length form distributive lattices under suitably defined partial orderings. We also give connections between the ternary period correlation of a partial word and its refined border correlation which specifies the lengths of all the word’s bordered cyclic shifts’ minimal borders. Finally, we investigate the population size, that is, the number of partial words sharing a given (refined) border correlation, and obtain formulas to compute it. We do so using the subgraph component polynomial of an undirected graph, introduced recently by Tittmann et al. (2011), which counts the number of connected components in vertex induced subgraphs.

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): F. Blanchet-Sadri, J. Lazarow, J. Nikkel, J.D. Quigley, X. Zhang This paper deals with two types of repetitions in strings: squares, which consist of two adjacent occurrences of substrings, and runs, which are periodic substrings that cannot be extended further to the left or right while maintaining the period. We show how to compute all the primitively-rooted squares in a given partial word, which is a sequence that may have undefined positions, called holes or wildcards, that match any letter of the alphabet over which the sequence is defined. We also describe an algorithm for computing all primitively-rooted runs in a given partial word and extend previous analyses on the number of runs to partial words.

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): Béla Bollobás, Shoham Letzter Given a word w of length n and i , j ∈ [ n ] , the longest common extension is the longest substring starting at both i and j . In this note we estimate the average length of the longest common extension over all words w and all pairs ( i , j ) , as well as the typical maximum length of the longest common extension. We also consider a variant of this problem, due to Blanchet-Sadri and Lazarow, in which the word is allowed to contain ‘holes’, which are special symbols functioning as ‘jokers’, i.e. are considered to be equal to any character. In particular, we estimate the average longest common extension over all words w with a small number of holes, extending a result by Blanchet-Sadri, Harred and Lazarow, and prove a similar result for words with holes appearing randomly.

Abstract: Publication date: February 2018 Source:European Journal of Combinatorics, Volume 68 Author(s): Mikhail Rubinchik, Arseny M. Shur We propose a new linear-size data structure which provides a fast access to all palindromic substrings of a string or a set of strings. This structure inherits some ideas from the construction of both the suffix trie and suffix tree. Using this structure, we present simple and efficient solutions for a number of problems involving palindromes.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): You Lu, Rong Luo, Cun-Quan Zhang For two pairs of vertices x 1 , y 1 and x 2 , y 2 , Seymour and Thomassen independently presented a characterization of graphs containing no edge-disjoint ( x 1 , y 1 ) -path and ( x 2 , y 2 ) -path. In this paper we first generalize their result to k ≥ 2 pairs of vertices. Namely, for 2 k vertices x 1 , y 1 , x 2 , y 2 , … , x k , y k , we characterize the graphs without edge-disjoint ( x i , y i ) -path and ( x j , y j ) -path for any 1 ≤ i < j ≤ k . Then applying this generalization, we present a characterization of signed graphs in which there are no edge-disjoint unbalanced circuits. Finally with this characterization we further show that every flow-admissible signed graph without edge-disjoint unbalanced circuits admits a nowhere-zero 6 -flow and thus verify the well-known Bouchet’s 6 -flow conjecture for this family of signed graphs.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Peter Frankl For two families of sets F , G ⊂ 2 [ n ] we define their set-wise union, F ∨ G = { F ∪ G : F ∈ F , G ∈ G } and establish several – hopefully useful – inequalities concerning F ∨ G . Some applications are provided as well.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Sarah J. Loeb, Douglas B. West The separation dimension of a graph G , written π ( G ) , is the minimum number of linear orderings of V ( G ) such that every two nonincident edges are “separated” in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the fractional separation dimension π f ( G ) , which is the minimum of a ∕ b such that some a linear orderings (repetition allowed) separate every two nonincident edges at least b times. In contrast to separation dimension, fractional separation dimension is bounded: always π f ( G ) ≤ 3 , with equality if and only if G contains K 4 . There is no stronger bound even for bipartite graphs, since π f ( K m , m ) = π f ( K m + 1 , m ) = 3 m m + 1 . We also compute π f ( G ) for cycles and some complete tripartite graphs. We show that π f ( G ) < 2 when G is a tree and present a sequence of trees on which the value tends to 4 ∕ 3 . Finally, we consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let π ∘ ( G ) be the number of circular orderings needed to separate all pairs and π f ∘ ( G ) be the fractional version. Among our results: (1) π ∘ ( G ) = 1 if and only G is outerplanar. (2) π ∘ ( G ) ≤ 2 when G is bipartite. (3) π ∘ ( K n ) ≥ log 2 log 3 ( n − 1 ) . (4) π f ∘ PubDate: 2017-10-10T03:40:15Z

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Gregory A. Freiman, Marcel Herzog, Patrizia Longobardi, Mercede Maj, Yonutz V. Stanchescu The aim of this paper is to present a complete description of the structure of finite subsets S of torsion-free nilpotent groups of class 2 satisfying S 2 = 3 S − 2 . In view of results in [12], this gives a complete description of the structure of finite subsets with the above property in any torsion-free nilpotent group.

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Yingli Kang The paper designs five graph operations, and proves that every signed graph with chromatic number q , defined by Kang and Steffen (in press), can be obtained from copies of the all-positive complete graph ( K q , + ) by repeatedly applying these operations. This result gives a signed version of the Hajós theorem, emphasizing the role of all-positive complete graphs in the class of signed graphs, as in the class of unsigned graphs. Moreover, a similar result is established for the signed chromatic number defined by Máčajová, Raspaud and Škoviera.

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Diego Nicodemos, Matěj Stehlík We show that any 3 -connected cubic plane graph on n vertices, with all faces of size at most 6 , can be made bipartite by deleting no more than ( p + 3 t ) n ∕ 5 edges, where p and t are the numbers of pentagonal and triangular faces, respectively. In particular, any such graph can be made bipartite by deleting at most 12 n ∕ 5 edges. This bound is tight, and we characterise the extremal graphs. We deduce tight lower bounds on the size of a maximum cut and a maximum independent set for this class of graphs. This extends and sharpens the results of Faria et al. (2012).

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Ron Aharoni, Dani Kotlar, Ran Ziv Drisko (1998) proved (essentially) that every family of 2 n − 1 matchings of size n in a bipartite graph possesses a partial rainbow matching of size n . In Barát et al. (2017) this was generalized as follows: Any ⌊ k + 2 k + 1 n ⌋ − ( k + 1 ) matchings of size n in a bipartite graph have a rainbow matching of size n − k . The paper has a twofold aim: (i) to extend these results to matchings of not necessarily equal cardinalities, and (ii) to prove a conjecture of Drisko, on the characterization of those families of 2 n − 2 matchings of size n in a bipartite graph that do not possess a rainbow matching of size n . Combining the latter with an idea of Alon (2011), we re-prove a characterization of the extreme case in a well-known theorem of Erdős–Ginzburg–Ziv in additive number theory.

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Emma Yu Jin We describe the thickened strips and introduce the outside nested decompositions of any skew shape λ ∕ μ . For any such decomposition Φ = ( Θ 1 , Θ 2 , … , Θ g ) of the skew shape λ ∕ μ where Θ i is a thickened strip for every i , let r be the number of boxes that are contained in any two distinct thickened strips of Φ . Then we establish a determinantal formula of the function p 1 r ( X ) s λ ∕ μ ( X ) with the Schur functions of thickened strips as entries, where s λ ∕ μ ( X ) is the Schur function of the skew shape λ ∕ μ and p 1 r ( X ) is the power sum symmetric function indexed by the partition ( 1 r ) . This generalizes Hamel and Goulden’s theorem on the outside decompositions of the skew shape λ ∕ μ and our extension is motivated by the enumeration of m -strip tableaux, which was first counted by Baryshnikov and Romik via extending the transfer operator approach due to Elkies.

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): G.R. Omidi, G. Raeisi, Z. Rahimi For given simple graphs G 1 , G 2 , … , G t , the Ramsey number R ( G 1 , G 2 , … , G t ) is the smallest positive integer n such that if the edges of the complete graph K n are partitioned into t disjoint color classes giving t graphs H 1 , H 2 , … , H t , then at least one H i has a subgraph isomorphic to G i . In this paper, for positive integers t 1 , t 2 , … , t s and n 1 , n 2 , … , n c the Ramsey number R ( S t 1 , S t 2 , … , S t s , n 1 K 2 , n 2 K 2 , … , n c K 2 ) is computed exactly, where n K 2 denotes a matching (stripe) of size n , i.e., n pairwise disjoint edges and S n is a star with n edges. This result generalizes and strengthens significantly a well-known result of Cockayne and Lorimer and also a known result of Gyárfás and Sárközy.

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): T. Kotek, J.A. Makowsky, E.V. Ravve Graph polynomials are deemed useful if they give rise to algebraic characterizations of various graph properties, and their evaluations encode many other graph invariants. Algebraic: The complete graphs K n and the complete bipartite graphs K n , n can be characterized as those graphs whose matching polynomials satisfy a certain recurrence relations and are related to the Hermite and Laguerre polynomials. An encoded graph invariant: The absolute value of the chromatic polynomial χ ( G , X ) of a graph G evaluated at − 1 counts the number of acyclic orientations of G . In this paper we prove a general theorem on graph families which are characterized by families of polynomials satisfying linear recurrence relations. This gives infinitely many instances similar to the characterization of K n , n . We also show where to use, instead of the Hermite and Laguerre polynomials, linear recurrence relations where the coefficients do not depend on n . Finally, we discuss the distinctive power of graph polynomials in specific form.

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): Karim Adiprasito We construct, for every dimension d ≥ 3 , polytopal spheres S for which neither S nor its dual S ∗ contain 2-faces that are triangles. This answers the topological formulation of a problem of Kalai, Kleinschmidt and Meisinger.

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Ilkyoo Choi, Jaehoon Kim, Alexandr V. Kostochka, André Raspaud A strong k -edge-coloring of a graph G is a mapping from E ( G ) to { 1 , 2 , … , k } such that every two adjacent edges or two edges adjacent to the same edge receive distinct colors. The strong chromatic index χ s ′ ( G ) of a graph G is the smallest integer k such that G admits a strong k -edge-coloring. We give bounds on χ s ′ ( G ) in terms of the maximum degree Δ ( G ) of a graph G when G is sparse, namely, when G is 2 -degenerate or when the maximum average degree Mad ( G ) is small. We prove that the strong chromatic index of each 2 -degenerate graph G is at most 5 Δ ( G ) + 1 . Furthermore, we show that for a graph G , if Mad ( G ) < 8 ∕ 3 and Δ ( G ) ≥ 9 , then χ s ′ ( G ) ≤ 3 Δ ( G ) − 3 (the bound 3 Δ ( G ) − 3 is sharp) and if Mad ( G ) < 3 and Δ ( G ) ≥ 7 , then χ s ′ ( G ) ≤ 3 Δ ( G ) (the restriction Mad ( G ) < 3 is sharp).

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Spencer Backman, Sam Hopkins, Lorenzo Traldi A fourientation of a graph G is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. We may naturally view fourientations as a mixture of subgraphs and graph orientations where unoriented and bioriented edges play the role of absent and present subgraph edges, respectively. Building on work of Backman and Hopkins (forthcoming), we show that given a linear order and a reference orientation of the edge set, one can define activities for fourientations of G which allow for a new 12 variable expansion of the Tutte polynomial T G . Our formula specializes to both an orientation activities expansion of T G due to Las Vergnas (1984) and a generalized activities expansion of T G due to Gordon and Traldi (1990).

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Guixin Deng, Xiangneng Zeng Let G be an abelian group (written additively), X be a subset of G and S be a minimal zero-sum sequence over X . S is called unsplittable in X if there do not exist an element g in S and two elements x , y in X such that g = x + y and the new sequence S g − 1 x y is still a minimal zero-sum sequence. In this paper, we mainly investigate the case when G = Z and X = 〚 − m , n 〛 with m , n ∈ N . We obtain the structure of unsplittable minimal zero-sum sequences of length at least n + ⌊ m ∕ 2 ⌋ + 2 provided that n ≥ m 2 ∕ 2 − 1 and m ≥ 6 . As a corollary, the Davenport constant D ( 〚 − m , n 〛 ) is determined when n ≥ m 2 ∕ 2 − 1 . The Davenport constant D ( X ) for a general set X ⊂ Z is also discussed.

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Erik Thörnblad The theory of tournament limits and tournament kernels is developed by extending common notions for finite tournaments to this setting; in particular we study transitivity and irreducibility of limits and kernels. We prove that each tournament kernel and each tournament limit can be decomposed into a direct sum of irreducible components, with transitive components interlaced. We also show that this decomposition is essentially unique.

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Nathan Bowler, Johannes Carmesin, Luke Postle We show that any infinite matroid can be reconstructed from the torsos of a tree-decomposition over its 2-separations, together with local information at the ends of the tree. We show that if the matroid is tame then this local information is simply a choice of whether circuits are permitted to use that end. The same is true if each torso is planar, with all gluing elements on a common face.

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Akiyoshi Tsuchiya It is known that a lattice simplex of dimension d corresponds a finite abelian subgroup of ( R ∕ Z ) d + 1 . Conversely, given a finite abelian subgroup of ( R ∕ Z ) d + 1 such that the sum of all entries of each element is an integer, we can obtain a lattice simplex of dimension d . In this paper, we discuss a characterization of Gorenstein simplices in terms of the associated finite abelian groups. In particular, we present complete characterizations of Gorenstein simplices whose normalized volume equals p , p 2 and p q , where p and q are prime numbers with p ≠ q . Moreover, we compute the volume of the associated dual reflexive simplices of the Gorenstein simplices.

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Hugo Parlier, Lionel Pournin We study flip-graphs of triangulations on topological surfaces where distance is measured by counting the number of necessary flip operations between two triangulations. We focus on surfaces of positive genus g with a single boundary curve and n marked points on this curve and consider triangulations up to homeomorphism with the marked points as their vertices. Our results are bounds on the maximal distance between two triangulations. Our lower bounds assert that these distances grow at least like 5 n ∕ 2 for all g ≥ 1 . Our upper bounds grow at most like [ 4 − 1 ∕ ( 4 g ) ] n for g ≥ 2 , and at most like 23 n ∕ 8 for the bordered torus.

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Márton Naszódi, Alexandr Polyanskii Johnson and Lovász and Stein proved independently that any hypergraph satisfies τ ≤ ( 1 + ln Δ ) τ ∗ , where τ is the transversal number, τ ∗ is its fractional version, and Δ denotes the maximum degree. We prove τ f ≤ 3 . 153 τ ∗ max { ln Δ , f } for the f -fold transversal number τ f . Similarly to Johnson, Lovász and Stein, we also show that this bound can be achieved non-probabilistically, using a greedy algorithm. As a combinatorial application, we prove an estimate on how fast τ f ∕ f converges to τ ∗ . As a geometric application, we obtain an upper bound on the minimal density of an f -fold covering of the d -dimensional Euclidean space by translates of any convex body.

Abstract: Publication date: January 2018 Source:European Journal of Combinatorics, Volume 67 Author(s): Maria Axenovich, Daniel Gonçalves, Jonathan Rollin, Torsten Ueckerdt The induced arboricity of a graph G is the smallest number of induced forests covering the edges of G . This is a well-defined parameter bounded from above by the number of edges of G when each forest in a cover consists of exactly one edge. Not all edges of a graph necessarily belong to induced forests with larger components. For k ⩾ 1 , we call an edge k -valid if it is contained in an induced tree on k edges. The k -strong induced arboricity of G , denoted by f k ( G ) , is the smallest number of induced forests with components of sizes at least k that cover all k -valid edges in G . This parameter is highly non-monotone. However, we prove that for any proper minor-closed graph class C , and more generally for any class of bounded expansion, and any k ⩾ 1 , the maximum value of f k ( G ) for G ∈ C is bounded from above by a constant depending only on C and k . This implies that the adjacent closed vertex-distinguishing number of graphs from a class of bounded expansion is bounded by a constant depending only on the class. We further prove that f 2 ( G ) ⩽ 3 t + 1 3 for any graph G of tree-width t and that f k ( G ) ⩽ ( 2 k ) d for any graph of tree-depth d . In addition, we prove that f 2 ( G ) ⩽ 310 when G is planar.