Abstract: Publication date: May 2018 Source:European Journal of Combinatorics, Volume 70 Author(s): Wenjie Fang We present a direct bijection between planar 3-connected triangulations and bridgeless planar maps, which were first enumerated by Tutte (1962) and Walsh and Lehman (1975) respectively. Previously known bijections by Wormald (1980) and Fusy (2010) are all defined recursively. Our direct bijection passes by a new class of combinatorial objects called “sticky trees”. We also present bijections between sticky trees, intervals in the Tamari lattices and closed flows on forests. With our bijections, we recover several known enumerative results about these objects. We thus show that sticky trees can serve as a nexus of bijective links among all these equi-enumerated objects.

Abstract: Publication date: May 2018 Source:European Journal of Combinatorics, Volume 70 Author(s): Victor J.W. Guo In an approach to the Cayley formula for counting trees, Shor discovered a refined recurrence relation concerning the number of improper edges. Chen and the author gave a bijection for the Shor recurrence based on the combinatorial interpretations of Zeng, answering a question of Shor. In this paper, we present a new bijective proof of the Shor recurrence by applying Shor’s formula for counting forests of rooted trees with roots 1 , … , r and with a given number of improper edges.

Abstract: Publication date: May 2018 Source:European Journal of Combinatorics, Volume 70 Author(s): Dennis Clemens, Shagnik Das, Tuan Tran The typical extremal problem asks how large a structure can be without containing a forbidden substructure. The Erdős–Rothschild problem, introduced in 1974 by Erdős and Rothschild in the context of extremal graph theory, is a coloured extension, asking for the maximum number of colourings a structure can have that avoid monochromatic copies of the forbidden substructure. The celebrated Erdős–Ko–Rado theorem is a fundamental result in extremal set theory, bounding the size of set families without a pair of disjoint sets, and has since been extended to several other discrete settings. The Erdős–Rothschild extensions of these theorems have also been studied in recent years, most notably by Hoppen, Koyakayawa and Lefmann for set families, and Hoppen, Lefmann and Odermann for vector spaces. In this paper we present a unified approach to the Erdős–Rothschild problem for intersecting structures, which allows us to extend the previous results, often with sharp bounds on the size of the ground set in terms of the other parameters. In many cases we also characterise which families of vector spaces asymptotically maximise the number of Erdős–Rothschild colourings, thus addressing a conjecture of Hoppen, Lefmann and Odermann.

Abstract: Publication date: May 2018 Source:European Journal of Combinatorics, Volume 70 Author(s): Béla Bollobás, Jonathan Lee, Shoham Letzter We consider the problem of maximising the largest eigenvalue of subgraphs of the hypercube Q d of a given order. We believe that in most cases, Hamming balls are maximisers, and our results support this belief. We show that the Hamming balls of radius o ( d ) have largest eigenvalue that is within 1 + o ( 1 ) of the maximum value. We also prove that Hamming balls with fixed radius maximise the largest eigenvalue exactly, rather than asymptotically, when d is sufficiently large. Our proofs rely on the method of compressions.

Abstract: Publication date: May 2018 Source:European Journal of Combinatorics, Volume 70 Author(s): Davide Bolognini, Antonio Macchia, Francesco Strazzanti Binomial edge ideals are a noteworthy class of binomial ideals that can be associated with graphs, generalizing the ideals of 2-minors. For bipartite graphs we prove the converse of Hartshorne’s Connectedness Theorem, according to which if an ideal is Cohen–Macaulay, then its dual graph is connected. This allows us to classify Cohen–Macaulay binomial edge ideals of bipartite graphs, giving an explicit and recursive construction in graph-theoretical terms. This result represents a binomial analogue of the celebrated characterization of (monomial) edge ideals of bipartite graphs due to Herzog and Hibi (2005).

Abstract: Publication date: May 2018 Source:European Journal of Combinatorics, Volume 70 Author(s): Alejandro H. Morales, Igor Pak, Greta Panova We give new bounds and asymptotic estimates on the number of standard Young tableaux of skew shape in a variety of special cases. Our approach is based on Naruse’s hook-length formula. We also compare our bounds with the existing bounds on the numbers of linear extensions of the corresponding posets.

Abstract: Publication date: May 2018 Source:European Journal of Combinatorics, Volume 70 Author(s): S.D. Adhikari, L. Boza, S. Eliahou, M.P. Revuelta, M.I. Sanz We show that, for every positive integer r , there exists an integer b = b ( r ) such that the 4-variable quadratic Diophantine equation ( x 1 − y 1 ) ( x 2 − y 2 ) = b is r -regular. Our proof uses Szemerédi’s theorem on arithmetic progressions.

Abstract: Publication date: May 2018 Source:European Journal of Combinatorics, Volume 70 Author(s): S.V. Konyagin, I.D. Shkredov We prove that asymptotically almost surely, the random Cayley sum graph over a finite abelian group G has edge density close to the expected one on every induced subgraph of size at least log c G , for any fixed c > 1 and G large enough.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): João Paulo Costalonga We establish the following splitter theorem for graphs and its generalization for matroids: Let G and H be 3 -connected simple graphs such that G has an H -minor and k ≔ V ( G ) − V ( H ) ≥ 2 . Let n ≔ k ∕ 2 + 1 . Then there are pairwise disjoint sets X 1 , … , X n ⊆ E ( G ) such that each G ∕ X i is a 3 -connected graph with an H -minor, each X i is a singleton set or the edge set of a triangle of G with 3 degree- 3 vertices and X 1 ∪ ⋯ ∪ X n contains no edge sets of circuits of G other than the X i ’s. This result extends previous ones of Whittle (for k = 1 , 2 ) and Costalonga (for k = 3 ).

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Zdeněk Dvořák We prove that for every graph H and for every integer s , the class of graphs that do not contain K s , K s , s , or any subdivision of H as induced subgraphs has bounded expansion; this strengthens a result of Kühn and Osthus (2004). The argument also gives another characterization of graph classes with bounded expansion and of nowhere-dense graph classes.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Daryl Funk, Dillon Mayhew, Steven D. Noble We give upper and lower bounds on the number of delta-matroids, and on the number of even delta-matroids.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Ferenc Szöllősi, Patric R.J. Östergård In this paper Seidel matrices are studied, and their spectrum and several related algebraic properties are determined for order n ≤ 13 . Based on this Seidel matrices with exactly three distinct eigenvalues of order n ≤ 23 are classified. One consequence of the computational results is that the maximum number of equiangular lines in R 12 with common angle 1 ∕ 5 is exactly 20 .

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Jakub Przybyło We consider the following extension of the concept of adjacent strong edge colourings of graphs without isolated edges. Two distinct vertices which are at distance at most r in a graph are called r -adjacent. The least number of colours in a proper edge colouring of a graph G such that the sets of colours met by any r -adjacent vertices in G are distinct is called the r -adjacent strong chromatic index of G and denoted by χ a , r ′ ( G ) . It has been conjectured that χ a , 1 ′ ( G ) ≤ Δ + 2 if G is connected of maximum degree Δ ≥ 2 and non-isomorphic to C 5 , while Hatami proved that there is a constant C , C ≤ 300 , such that χ a , 1 ′ ( G ) ≤ Δ + C if Δ > 1 0 20 [J. Combin. Theory Ser. B 95 (2005) 246–256]. We conjecture that a similar statement should hold for any r , i.e., that for each positive integer r there exist constants δ 0 and C such that χ a , r ′ ( G ) ≤ Δ + C for every graph without an isolated edge and with minimum degree δ ≥ δ 0 , and argue that a lower bound on δ is unavoidable in such a case (for r > 2 ). Using the probabilistic method we prove such an upper bound to hold for graphs with δ ≥ ϵ Δ , for every r and any fixed ε ∈ ( 0 , 1 ] , i.e., in particular for regular graphs. We also support the conjecture by proving an upper bound χ a , r ′ ( G ) ≤ ( 1 + o ( 1 ) ) Δ for graphs with δ ≥ r + 2 .

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Marthe Bonamy, Nicolas Bousquet Let k be an integer. Two vertex k -colorings of a graph are adjacent if they differ on exactly one vertex. A graph is k -mixing if any proper k -coloring can be transformed into any other through a sequence of adjacent proper k -colorings. Jerrum proved that any graph is k -mixing if k is at least the maximum degree plus two. We first improve Jerrum’s bound using the Grundy number, which is the greatest number of colors in a greedy coloring. As a corollary, we obtain that any cograph G is ( χ ( G ) + 1 ) -mixing, and that for fixed χ ( G ) the shortest sequence between any two ( χ ( G ) + 1 ) -colorings is at most linear in the number of vertices. We additionally argue that while cographs are exactly the P 4 -free graphs, the result cannot be extended to P 5 -free graphs. Any graph is ( tw + 2 ) -mixing, where tw denotes the treewidth of the graph (Cereceda 2006). We prove that the shortest sequence between any two ( tw + 2 ) -colorings is at most quadratic (which is optimal up to a constant factor), a problem left open in Bonamy et al. (2012). All the proofs are constructive and lead to polynomial-time recoloring algorithms: given two colorings, we can exhibit in polynomial time a sequence transforming one into the other.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Tim Römer, Sara Saeedi Madani Given a graph G , the cut polytope is the convex hull of its cut vectors. The latter objects are the incidence vectors associated to all cuts of G . Especially motivated by related conjectures of Sturmfels and Sullivant, we study various properties and invariants of the toric algebra of the cut polytope, called its cut algebra. In particular, we characterize those cut algebras which are complete intersections, have linear resolutions or have Castelnuovo–Mumford regularity equal to 2. The key idea of our approach is to consider suitable algebra retracts of cut algebras. Additionally, combinatorial retracts of the graph are defined and investigated, which are special minors whose algebraic properties can be compared in a very pleasant way with the corresponding ones of the original graph. Moreover, we discuss several examples and pose new problems as well.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Benjamin Braun, McCabe Olsen We consider quotients of the unit cube semigroup algebra by particular Z r ≀ S n -invariant ideals. Using Gröbner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by colored permutations ( π , ϵ ) ∈ Z r ≀ S n and each element encodes the negative descent and negative major index statistics on ( π , ϵ ) . This gives an algebraic interpretation of these statistics that was previously unknown. This basis of the Z r ≀ S n -quotients allows us to recover certain combinatorial identities involving Euler–Mahonian distributions of statistics.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Jakub Sosnovec We study the dispersion of a point set, a notion closely related to the discrepancy. Given a real r ∈ ( 0 , 1 ) and an integer d ≥ 2 , let N ( r , d ) denote the minimum number of points inside the d -dimensional unit cube [ 0 , 1 ] d such that they intersect every axis-aligned box inside [ 0 , 1 ] d of volume greater than r . We prove an upper bound on N ( r , d ) , matching a lower bound of Aistleitner et al. up to a multiplicative constant depending only on r . This fully determines the rate of growth of N ( r , d ) if r ∈ ( 0 , 1 ) is fixed.

Abstract: Publication date: March 2018 Source:European Journal of Combinatorics, Volume 69 Author(s): Torsten Mütze, Christoph Standke, Veit Wiechert A Dyck path of length 2 k with e flaws is a path in the integer lattice that starts at the origin and consists of k many ↗ -steps and k many ↘ -steps that change the current coordinate by ( 1 , 1 ) or ( 1 , − 1 ) , respectively, and that has exactly e many ↘ -steps below the line y = 0 . Denoting by D 2 k e the set of Dyck paths of length 2 k with e flaws, the well-known Chung–Feller theorem asserts that the sets D 2 k 0 , D 2 k 1 , … , D 2 k k all have the same cardinality 1 k + 1 2 k k = C k , the k th Catalan number. The standard combinatorial proof of this classical result establishes a bijection f ′ between D 2 k e and D 2 k e + 1 that swaps certain parts of the given Dyck path x , with the effect that x and f ′ ( x ) may differ in many positions. In this paper we strengthen the Chung–Feller theorem by presenting a simple bijection f between D 2 k e and D 2 k e + 1 which has the additional feature that x and f ( x ) differ in only two positions (the least possible number). We also present an algorithm that allows to compute a sequence of applications of f in constant time per generated Dyck path. As an application, we use our minimum-change bijection f to construct cycle-factors in the odd graph O 2 k + 1 and the middle levels graph PubDate: 2017-12-26T18:10:18Z

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): 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): 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.