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: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): Martin Balko, Pavel Valtr A classical conjecture of Erdős and Szekeres states that, for every integer k ≥ 2 , every set of 2 k − 2 + 1 points in the plane in general position contains k points in convex position. In 2006, Peters and Szekeres introduced the following stronger conjecture: every red-blue coloring of the edges of the ordered complete 3-uniform hypergraph on 2 k − 2 + 1 vertices contains an ordered subhypergraph with k vertices and k − 2 edges, which is a union of a red monotone path and a blue monotone path that are vertex disjoint except for their two common end-vertices. Applying a state of art SAT solver, we refute the conjecture of Peters and Szekeres. We also apply techniques of Erdős, Tuza, and Valtr to refine the Erdős–Szekeres conjecture in order to tackle it with SAT solvers.

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): F. Botler, G.O. Mota, M.T.I. Oshiro, Y. Wakabayashi A P ℓ -decomposition of a graph G is a set of pairwise edge-disjoint paths with ℓ edges that cover the edge set of G . In 1957, Kotzig proved that a 3 -regular graph admits a P 3 -decomposition if and only if it contains a perfect matching, and also asked what are the necessary and sufficient conditions for an ℓ -regular graph to admit a P ℓ -decomposition, for odd ℓ . Let g , ℓ and m be positive integers with g ≥ 3 . We prove that, (i) if ℓ is odd and m > 2 ⌊ ( ℓ − 2 ) ∕ ( g − 2 ) ⌋ , then every m ℓ -regular graph with girth at least g that contains an m -factor admits a P ℓ -decomposition; (ii) if m > ⌊ ( ℓ − 2 ) ∕ ( g − 2 ) ⌋ , then every 2 m ℓ -regular graph with girth at least g admits a P ℓ -decomposition. Furthermore, we prove that, for graphs with girth at least ℓ − 1 , statement (i) holds for every m ≥ 1 ; and observe that, statement (ii) also holds for every m ≥ 1 .

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): Michał Dębski, Zbigniew Lonc, Paweł Rzążewski A harmonious coloring of a k -uniform hypergraph H is a rainbow vertex coloring such that each k -set of colors appears on at most one edge. A rainbow coloring of H is achromatic if each k -set of colors appears on at least one edge. The harmonious number (resp. achromatic number) of H , denoted by h ( H ) (resp. ψ ( H ) ) is the minimum (resp. maximum) possible number of colors in a harmonious (resp. achromatic) coloring of H . A class H of hypergraphs is fragmentable if for every H ∈ H , H can be fragmented into components of a bounded size by removing a “small” fraction of vertices. We show that for every fragmentable class H of bounded degree hypergraphs, for every ϵ > 0 and for every hypergraph H ∈ H with m ≥ m 0 ( H , ϵ ) edges we have h ( H ) ≤ ( 1 + ϵ ) k ! m k and ψ ( H ) ≥ ( 1 − ϵ ) k ! m k . As corollaries, we answer a question posed by Blackburn concerning the maximum length of t -subset packing sequences of constant radius and derive an asymptotically tight bound on the minimum number of colors in a vertex-distinguishing edge coloring of cubic planar graphs, which is a step towards confirming a conjecture of Burris and Schelp.

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): François Dross, Mickael Montassier, Alexandre Pinlou An ( F , F d ) -partition of a graph is a vertex-partition into two sets F and F d such that the graph induced by F is a forest and the one induced by F d is a forest with maximum degree at most d . We prove that every triangle-free planar graph admits an ( F , F 5 ) -partition. Moreover we show that if for some integer d there exists a triangle-free planar graph that does not admit an ( F , F d ) -partition, then it is an NP-complete problem to decide whether a triangle-free planar graph admits such a partition.

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): Zdeněk Dvořák, Ken-ichi Kawarabayashi We prove that every triangle-free graph of tree-width t has chromatic number at most ⌈ ( t + 3 ) ∕ 2 ⌉ , and demonstrate that this bound is tight. The argument also establishes a connection between coloring graphs of tree-width t and on-line coloring of graphs of path-width t .

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): Jan van den Heuvel, Patrice Ossona de Mendez, Daniel Quiroz, Roman Rabinovich, Sebastian Siebertz The generalised colouring numbers col r ( G ) and wcol r ( G ) were introduced by Kierstead and Yang as a generalisation of the usual colouring number, and have since then found important theoretical and algorithmic applications. In this paper, we dramatically improve upon the known upper bounds for generalised colouring numbers for graphs excluding a fixed minor, from the exponential bounds of Grohe et al. to a linear bound for the r -colouring number col r and a polynomial bound for the weak r -colouring number wcol r . In particular, we show that if G excludes K t as a minor, for some fixed t ≥ 4 , then col r ( G ) ≤ t − 1 2 ( 2 r + 1 ) and wcol r ( G ) ≤ r + t − 2 t − 2 ( t − 3 ) ( 2 r + 1 ) ∈ O ( r t − 1 ) . In the case of graphs G of bounded genus g , we improve the bounds to col r ( G ) ≤ ( 2 g + 3 ) ( 2 r + 1 ) (and even col r ( G ) ≤ 5 r + 1 if g = 0 , i.e. if G is planar) and wcol r ( G ) ≤ ( 2 g + r + 2 2 ) ( 2 r + 1 ) .

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): Lucas Hosseini, Jaroslav Nešetřil, Patrice Ossona de Mendez In this paper we consider a simple algebraic structure — sets with a single endofunction. We shall see that from the point of view of structural limits, even this simplest case is both interesting and difficult. Nevertheless we obtain the shape of limit objects in the full generality, and we prove the inverse theorem in the case of quantifier-free limits.

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): Winfried Hochstättler We transfer Tutte’s theory for analyzing the chromatic number of a graph using nowhere-zero-coflows and -flows (NZ-flows) to the dichromatic number of a digraph and define Neumann-Lara-flows (NL-flows). We prove that any digraph whose underlying (multi-)graph is 3 -edge-connected admits a NL-3-flow, and even a NL-2-flow in case the underlying graph is 4 -edge connected. We conjecture that 3 -edge-connectivity already guarantees the existence of a NL-2-flow, which, if true, would imply the 2-Color-Conjecture for planar graphs due to Víctor Neumann-Lara. Finally we present an extension of the theory to oriented matroids.

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): Carlos Hoppen, Hanno Lefmann, Knut Odermann Inspired by previous work of Balogh (2006), we show that, given r ≥ 5 and n large, the balanced complete bipartite graph K n ∕ 2 , n ∕ 2 is the n -vertex graph that admits the largest number of r -edge-colorings for which there is no triangle whose edges are assigned three distinct colors. Moreover, we extend this result to lower values of n when r ≥ 10 , and we provide approximate results for r ∈ { 3 , 4 } .

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): Lars Jaffke, Hans L. Bodlaender, Pinar Heggernes, Jan Arne Telle One of the most famous algorithmic meta-theorems states that every graph property which can be defined in counting monadic second order logic (CMSOL) can be checked in linear time on graphs of bounded treewidth, which is known as Courcelle’s Theorem (Courcelle, 1990). These algorithms are constructed as finite state tree automata and hence every CMSOL-definable graph property is recognizable. Courcelle also conjectured that the converse holds, i.e. every recognizable graph property is definable in CMSOL for graphs of bounded treewidth. In this paper we prove two special cases of this conjecture, first for the class of k -outerplanar graphs, which are known to have treewidth at most 3 k − 1 (Bodlaender, 1998) and for graphs of bounded treewidth without chordless cycles of length at least some constant ℓ . We furthermore show that for a proof of Courcelle’s Conjecture it is sufficient to show that all members of a graph class admit constant width tree decompositions whose bags and edges can be identified with MSOL-predicates. For graph classes that admit MSOL-definable constant width tree decompositions that have bounded degree or allow for a linear ordering of all nodes with the same parent we even give a stronger result: In that case, the counting predicates of CMSOL are not needed.

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): Nina Kamčev, Benny Sudakov, Jan Volec Let c be an edge-coloring of the complete n -vertex graph K n . The problem of finding properly colored and rainbow Hamilton cycles in c was initiated in 1976 by Bollobás and Erdős and has been extensively studied since then. Recently it was extended to the hypergraph setting by Dudek et al. (2012). We generalize these results, giving sufficient local (resp. global) restrictions on the colorings which guarantee a properly colored (resp. rainbow) copy of a given hypergraph G . We also study multipartite analogues of these questions. We give (up to a constant factor) optimal sufficient conditions for a coloring c of the complete balanced m -partite graph to contain a properly colored or rainbow copy of a given graph G with maximum degree Δ . Our bounds exhibit a surprising transition in the rate of growth, showing that the problem is fundamentally different in the regimes Δ ≫ m and Δ ≪ m . Our main tool is the framework of Lu and Székely for the space of random bijections, which we extend to product spaces.

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): Robert Morris Many fundamental and important questions from statistical physics lead to beautiful problems in extremal and probabilistic combinatorics. One particular example of this phenomenon is the study of bootstrap percolation, which is motivated by a variety of ‘real-world’ cellular automata, such as the Glauber dynamics of the Ising model of ferromagnetism, and kinetically constrained spin models of the liquid–glass transition. In this review article, we will describe some dramatic recent developments in the theory of bootstrap percolation (and, more generally, of monotone cellular automata with random initial conditions), and discuss some potential extensions of these methods and results to other automata. In particular, we will state numerous conjectures and open problems.

Abstract: Publication date: December 2017 Source:European Journal of Combinatorics, Volume 66 Author(s): Juanjo Rué, Oriol Serra, Lluis Vena We provide asymptotic counting for the number of subsets of given size which are free of certain configurations in finite groups. Applications include sets without solutions to equations in non-abelian groups, and linear configurations in abelian groups defined from group homomorphisms. The results are obtained by combining the methodology of hypergraph containers joint with arithmetic removal lemmas. Random sparse versions and threshold probabilities for existence of configurations in sets of given density are presented as well.

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: Available online 12 August 2017 Source:European Journal of Combinatorics Author(s): Jiří Fiala, Jan Hubička, Yangjing Long, Jaroslav Nešetřil We show that every interval in the homomorphism order of finite undirected graphs is either universal or a gap. Together with density and universality this “fractal” property contributes to the spectacular properties of the homomorphism order. We first show the fractal property by using Sparse Incomparability Lemma and then by a more involved elementary argument.

Abstract: Publication date: Available online 9 August 2017 Source:European Journal of Combinatorics Author(s): Luke Postle Kostochka and Yancey proved that every 5 -critical graph G satisfies: E ( G ) ≥ 9 4 V ( G ) − 5 4 . A construction of Ore gives an infinite family of graphs meeting this bound. We prove that there exists ϵ , δ > 0 such that if G is a 5 -critical graph, then E ( G ) ≥ ( 9 4 + ϵ ) V ( G ) − 5 4 − δ T ( G ) where T ( G ) is the maximum number of vertex-disjoint cliques of size three or four where cliques of size four have twice the weight of a clique of size three. As a corollary, a triangle-free 5 -critical graph G satisfies: E ( G ) ≥ ( 9 4 + ϵ ) V ( G ) − 5 4 .

Abstract: Publication date: Available online 7 August 2017 Source:European Journal of Combinatorics Author(s): Robert Hancock, Andrew Treglown Given a linear equation L , a set A ⊆ [ n ] is L -free if A does not contain any ‘non-trivial’ solutions to L . In this paper we consider the following three general questions: (i) What is the size of the largest L -free subset of [ n ] ' (ii) How many L -free subsets of [ n ] are there' (iii) How many maximal L -free subsets of [ n ] are there' We completely resolve (i) in the case when L is the equation p x + q y = z for fixed p , q ∈ N where p ≥ 2 . Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations L , thereby refining a special case of a result of Green (2005). We also give various bounds on the number of maximal L -free subsets of [ n ] for three-variable homogeneous linear equations L . For this, we make use of container and removal lemmas of Green (2005).

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.

Abstract: Publication date: October 2017 Source:European Journal of Combinatorics, Volume 65 Author(s): Tony Huynh, Sang-il Oum, Maryam Verdian-Rizi An even-cycle decomposition of a graph G is a partition of E ( G ) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no K 5 -minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2-connected loopless Eulerian odd- K 4 -minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that ‘odd- K 4 -minor-free’ cannot be replaced with ‘odd- K 5 -minor-free.’ The main technical ingredient is a structural characterization of the class of odd- K 4 -minor-free graphs, which is due to Lovász, Seymour, Schrijver, and Truemper.

Abstract: Publication date: October 2017 Source:European Journal of Combinatorics, Volume 65 Author(s): Sizhong Zhou, Zhiren Sun, Zurun Xu Let G be a digraph with vertex set V ( G ) and arc set E ( G ) . Let m , r , k be three positive integers, and let f = ( f − , f + ) be a pair of nonnegative integer-valued functions defined on V ( G ) with f ( x ) ≥ ( k + 1 ) r for all x ∈ V ( G ) . Let H 1 , H 2 , … , H k be k vertex disjoint m r -subdigraphs of G . In this paper, it is proved that every ( 0 , m f − ( m − 1 ) r ) -digraph has a ( 0 , f ) -factorization r -orthogonal to every H i ( i = 1 , 2 , … , k ).

Abstract: Publication date: October 2017 Source:European Journal of Combinatorics, Volume 65 Author(s): S.V. Avgustinovich, A.E. Frid, S. Puzynina An infinite permutation is a linear ordering of the set of natural numbers. An infinite permutation can be defined by a sequence of real numbers where only the order of elements is taken into account; such sequence of reals is called a representative of a permutation. In this paper we consider infinite permutations which possess an equidistributed representative on [ 0 , 1 ] (i.e., such that the prefix frequency of elements from each interval exists and is equal to the length of this interval), and we call such permutations equidistributed. Similarly to infinite words, a complexity p ( n ) of an infinite permutation is defined as a function counting the number of its subpermutations of length n . We show that, unlike for permutations in general, the minimal complexity of an equidistributed permutation α is p α ( n ) = n , establishing an analog of Morse and Hedlund theorem. The class of equidistributed permutations of minimal complexity coincides with the class of so-called Sturmian permutations, directly related to Sturmian words.

Abstract: Publication date: October 2017 Source:European Journal of Combinatorics, Volume 65 Author(s): Zoltán L. Blázsik, Zoltán Lóránt Nagy We determine the partition dimension of the incidence graph G ( Π q ) of the projective plane Π q up to a constant factor 2 as ( 2 + o ( 1 ) ) log 2 q ≤ pd ( G ( Π q ) ) ≤ ( 4 + o ( 1 ) ) log 2 q .

Abstract: Publication date: October 2017 Source:European Journal of Combinatorics, Volume 65 Author(s): Haimiao Chen Given a finite group Γ , a regular t -balanced Cayley map ( RBCM t for short) is a regular Cayley map CM ( G , Ω , ρ ) such that ρ ( ω ) − 1 = ρ t ( ω ) for all ω ∈ Ω . In this paper, we clarify a connection between quotients of polynomial rings and RBCM t ’s on abelian groups, so as to propose a new approach for classifying RBCM t ’s. We obtain many new results, in particular, a complete classification for RBCM t ’s on abelian 2-groups.

Abstract: Publication date: October 2017 Source:European Journal of Combinatorics, Volume 65 Author(s): Nicolas Bonichon, Mireille Bousquet-Mélou, Paul Dorbec, Claire Pennarun The number of planar Eulerian maps with n edges is well-known to have a simple expression. But what is the number of planar Eulerian orientations with n edges' This problem appears to be a difficult one. To approach it, we define and count families of subsets and supersets of planar Eulerian orientations, indexed by an integer k , that converge to the set of all planar Eulerian orientations as k increases. The generating functions of our subsets can be characterized by systems of polynomial equations, and are thus algebraic. The generating functions of our supersets are characterized by polynomial systems involving divided differences, as often occurs in map enumeration. We prove that these series are algebraic as well. We obtain in this way lower and upper bounds on the growth rate of planar Eulerian orientations, which appears to be around 12.5.