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Kernelization and approximation of distance-r independent sets on nowhere dense graphs Eur. J. Comb. (IF 0.848) Pub Date : 2021-01-22 Michał Pilipczuk; Sebastian Siebertz
For a positive integer r, a distance-r independent set in an undirected graph G is a set I⊆V(G) of vertices pairwise at distance greater than r, while a distance-r dominating set is a set D⊆V(G) such that every vertex of the graph is within distance at most r from a vertex from D. We study the duality between the maximum size of a distance-2r independent set and the minimum size of a distance-r dominating
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Combinatorics on lattice paths in strips Eur. J. Comb. (IF 0.848) Pub Date : 2021-01-22 Nancy S.S. Gu; Helmut Prodinger
For lattice paths in strips which begin at (0,0) and have only up steps U:(i,j)→(i+1,j+1) and down steps D:(i,j)→(i+1,j−1), let An,k denote the set of paths of length n which start at (0,0), end on heights 0 or −1, and are contained in the strip −⌊k+12⌋≤y≤⌊k2⌋ of width k, and let Bn,k denote the set of paths of length n which start at (0,0) and are contained in the strip 0≤y≤k. We establish a bijection
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Rigid realizations of graphs with few locations in the plane Eur. J. Comb. (IF 0.848) Pub Date : 2021-01-14 Csaba Király
Adiprasito and Nevo (2018) proved that there exists a set of 76 points in R3 such that every triangulated planar graph has an infinitesimally rigid realization in which each vertex is mapped to a point in this set. In this paper we show that there exists a set of 26 points in the plane such that every planar graph which is generically rigid in R2 has an infinitesimally rigid realization in which each
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On the Standard (2,2)-Conjecture Eur. J. Comb. (IF 0.848) Pub Date : 2021-01-08 Jakub Przybyło
The well-known 1–2–3 Conjecture asserts that the edges of every graph without an isolated edge can be weighted with 1, 2 and 3 so that adjacent vertices receive distinct weighted degrees. This is open in general. The Standard (2,2)-Conjecture asserts that every graph with no isolated edge and no isolated triangle can be decomposed into two graphs, each of which can be weighted with 1, 2 for the same
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Remarks on the Erdős Matching Conjecture for vector spaces Eur. J. Comb. (IF 0.848) Pub Date : 2021-01-08 Ferdinand Ihringer
In 1965, Paul Erdős asked about the largest family Y of k-sets in {1,…,n} such that Y does not contain s+1 pairwise disjoint sets. This problem is commonly known as the Erdős Matching Conjecture. We investigate the q-analog of this question, that is we want to determine the size of a largest family Y of k-spaces in Fqn such that Y does not contain s+1 pairwise disjoint k-spaces. Here we call two subspaces
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Atoms for signed permutations Eur. J. Comb. (IF 0.848) Pub Date : 2021-01-06 Zachary Hamaker; Eric Marberg
There is a natural analogue of weak Bruhat order on the involutions in any Coxeter group. The saturated chains of intervals in this order correspond to reduced words for a certain set of group elements called atoms. Brion gives a general formula for the cohomology class of a K-orbit closure in an arbitrary flag variety, where K is a symmetric subgroup of a complex algebraic group. In type A, the terms
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Entire coloring of 2-connected plane graphs Eur. J. Comb. (IF 0.848) Pub Date : 2021-01-04 Weifan Wang; Xiaoxue Hu; Yiqiao Wang
The entire chromatic number χvef(G) of a plane graph G is the least number of colors such that any two adjacent or incident elements in V(G)∪E(G)∪F(G) receive different colors. In this paper, we prove that every 2-connected simple plane graph G with maximum degree Δ≥20 has χvef(G)=Δ+1.
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Cyclic coloring of plane graphs with maximum face size 16 and 17 Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-30 Zdeněk Dvořák; Michael Hebdige; Filip Hlásek; Daniel Král’; Jonathan A. Noel
Plummer and Toft conjectured in 1987 that the vertices of every 3-connected plane graph with maximum face size Δ⋆ can be colored using at most Δ⋆+2 colors in such a way that no face is incident with two vertices of the same color. The conjecture has been proven for Δ⋆=3, Δ⋆=4 and Δ⋆≥18. We prove the conjecture for Δ⋆=16 and Δ⋆=17.
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Connectivity of orientations of 3-edge-connected graphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-24 Florian Hörsch; Zoltán Szigeti
We attempt to generalize a theorem of Nash-Williams stating that a graph has a k-arc-connected orientation if and only if it is 2k-edge-connected. In a strongly connected digraph we call an arc deletable if its deletion leaves a strongly connected digraph. Given a 3-edge-connected graph G, we define its Frank number f(G) to be the minimum number k such that there exist k orientations of G with the
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Integer sets with identical representation functions, II Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-23 Shi-Qiang Chen; Yong-Gao Chen
Let N be the set of all nonnegative integers. For S⊆N and n∈N, let RS(n) denote the number of solutions of the equation n=s+s′, s,s′∈S, s
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Enumeration of various animals on the triangular lattice Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-22 Toufik Mansour; Reza Rastegar
In this paper, we consider various classes of polyiamonds that are animals residing on the triangular lattice. By careful analyses through certain layer-by-layer decompositions and cell pruning/growing arguments, we derive explicit forms for the generating functions of the number of nonempty translation-invariant baryiamonds (bargraphs in the triangular lattice), column-convex polyiamonds, and convex
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On Rado conditions for nonlinear Diophantine equations Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-17 Jordan Mitchell Barrett; Martino Lupini; Joel Moreira
Building on previous work of Di Nasso and Luperi Baglini, we provide general necessary conditions for a Diophantine equation to be partition regular. These conditions are inspired by Rado’s characterization of partition regular linear homogeneous equations. We conjecture that these conditions are also sufficient for partition regularity, at least for equations whose corresponding monovariate polynomial
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Principal specializations of Schubert polynomials and pattern containment Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-17 Yibo Gao
The principal specialization νw=Sw(1,…,1) of the Schubert polynomial at w, which equals the degree of the matrix Schubert variety corresponding to w, has attracted a lot of attention in recent years. In this paper, we show that νw is bounded below by 1+p132(w)+p1432(w) where pu(w) is the number of occurrences of the pattern u in w, strengthening a previous result by A. Weigandt. We then make a conjecture
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Determinantal and Pfaffian identities for ninth variation skew Schur functions and Q-functions Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-16 Angèle M. Foley; Ronald C. King
Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald’s ninth variation skew Schur functions. Here we introduce a skew shifted tableaux definition of these ninth variation skew Q-functions, and prove by means of a non-intersecting lattice path model a Pfaffian outside decomposition result in the form of a ninth variation version of Hamel’s Pfaffian outside
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Graph covers with two new eigenvalues Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-14 Chris Godsil; Maxwell Levit; Olha Silina
A certain signed adjacency matrix of the hypercube, which Hao Huang used last year to resolve the Sensitivity Conjecture, is closely related to the unique, 4-cycle free, 2-fold cover of the hypercube. We develop a framework in which this connection is a natural first example of the relationship between group valued adjacency matrices with few eigenvalues, and combinatorially interesting covering graphs
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A linear hypergraph extension of the bipartite Turán problem Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-15 Guorong Gao; An Chang
An r-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of r-uniform hypergraphs F, the linear Turán number exrlin(n,F) is the maximum number of edges of a linear r-uniform hypergraph on n vertices that does not contain any member of F as a subhypergraph. Given a graph F and a positive integer r≥2, the r-expansion of F is the r-graph F+ obtained from F by
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On 12̲0-avoiding inversion and ascent sequences Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-14 Zhicong Lin; Shishuo Fu
Recently, Yan and the first author investigated systematically the enumeration of inversion or ascent sequences avoiding vincular patterns of length 3, where two of the three letters are required to be adjacent. They established many connections with familiar combinatorial families and proposed several interesting conjectures. The main objective of this paper is to prove two of their conjectures concerning
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Defective DP-colorings of sparse multigraphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-14 Yifan Jing; Alexandr Kostochka; Fuhong Ma; Pongpat Sittitrai; Jingwei Xu
DP-coloring (also known as correspondence coloring) is a generalization of list coloring developed recently by Dvořák and Postle. We introduce and study (i,j)-defective DP-colorings of multigraphs. We concentrate on sparse multigraphs and consider fDP(i,j,n) — the minimum number of edges that may have an n-vertex (i,j)-critical multigraph, that is, a multigraph G that has no (i,j)-defective DP-coloring
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Tilings in graphons Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-11 Jan Hladký; Ping Hu; Diana Piguet
We introduce a counterpart to the notion of tilings, that is vertex-disjoint copies of a fixed graph F, to the setting of graphons. The case F=K2 gives the notion of matchings in graphons. We give a transference statement that allows us to switch between the finite and limit notion, and derive several favorable properties, including the LP-duality counterpart to the classical relation between the fractional
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Alternating multiple zeta values, and explicit formulas of some Euler–Apéry-type series Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-11 Weiping Wang; Ce Xu
In this paper, we study some Euler–Apéry-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to ln(2), zeta values, and alternating multiple zeta values by
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Lovász–Saks–Schrijver ideals and parity binomial edge ideals of graphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-11 Arvind Kumar
Let G be a simple graph on n vertices. Let LG and IG denote the Lovász–Saks–Schrijver(LSS) ideal and parity binomial edge ideal of G in the polynomial ring S=K[x1,…,xn,y1,…,yn] respectively. We classify graphs whose LSS ideals and parity binomial edge ideals are complete intersections. We also classify graphs whose LSS ideals and parity binomial edge ideals are almost complete intersections, and we
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Curves on surfaces and surgeries Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-11 Abdoul Karim Sane
In this paper, we introduce a new surgery operation on the set of collections of curves, on a closed oriented surface, whose complement is a topological disk. We prove that any two such collections can be connected by a sequence of surgeries.
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Fertility monotonicity and average complexity of the stack-sorting map Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-10 Colin Defant
Let Dn denote the average number of iterations of West’s stack-sorting map s that are needed to sort a permutation in Sn into the identity permutation 123⋯n. We prove that 0.62433≈λ≤lim infn→∞Dnn≤lim supn→∞Dnn≤35(7−8log2)≈0.87289, where λ is the Golomb–Dickman constant. Our lower bound improves upon West’s lower bound of 0.23, and our upper bound is the first improvement upon the trivial upper bound
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Analogues of Milner’s Theorem for families without long chains and of vector spaces Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-10 Peter Frankl
Let n>k>0 be integers and X an n-element set. A family F consisting of subsets of X is called k-Sperner if it has no distinct members F0,…,Fk such that F0⊂F1⊂…⊂Fk. A family is called s-union if the union of any two of its members has size at most s. A classical result of Milner determines the maximum size of a family that is both 1-Sperner and s-union. The present paper is dealing with the case k≥2
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On the asymptotic growth of the number of tree-child networks Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-10 Michael Fuchs; Guan-Ru Yu; Louxin Zhang
In a recent paper, McDiarmid, Semple, and Welsh (2015) showed that the number of tree-child networks with n leaves has the factor n2n in its main asymptotic growth term. In this paper, we improve this by completely identifying the main asymptotic growth term up to a constant. More precisely, we show that the number of tree-child networks with n leaves grows like Θn−2∕3ea1(3n)1∕312e2nn2n,where a1=−2
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Finite 3-set-homogeneous graphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-10 Jin-Xin Zhou
In this paper, all graphs are assumed to be finite. Let s≥1 be an integer. A graph is called s-CSH (s-connected-set-homogeneous) if for every pair of isomorphic connected induced subgraphs on at most s vertices, there exists an automorphism mapping the first to the second. A graph is called s-SH (s-set-homogeneous) if for every pair of isomorphic induced subgraphs (not necessarily connected) on at
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Representations of torsion-free arithmetic matroids Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-10 Roberto Pagaria; Giovanni Paolini
We study the representability problem for torsion-free arithmetic matroids. After introducing a “strong gcd property” and a new operation called “reduction”, we describe and implement an algorithm to compute all essential representations, up to equivalence. As a consequence, we obtain an upper bound to the number of equivalence classes of representations. In order to rule out equivalent representations
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Counterexamples to a conjecture by Gross, Mansour and Tucker on partial-dual genus polynomials of ribbon graphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-09 Qi Yan; Xian’an Jin
Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They computed these two partial-dual genus polynomials of four families of ribbon graphs, posed some research problems and made some conjectures. In this paper, we introduce the notion of signed interlace sequences of bouquets and obtain the partial-dual Euler genus polynomials
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A note on sublinear separators and expansion Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-04 Zdeněk Dvořák
For a hereditary class G of graphs, let sG(n) be the minimum function such that each n-vertex graph in G has a balanced separator of order at most sG(n), and let ∇G(r) be the minimum function bounding the expansion of G, in the sense of bounded expansion theory of Nešetřil and Ossona de Mendez. The results of Plotkin et al. (1994) and Esperet and Raymond (2018) imply that if sG(n)=Θ(n1−ε) for some
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Trees, parking functions and factorizations of full cycles Eur. J. Comb. (IF 0.848) Pub Date : 2020-12-02 John Irving; Amarpreet Rattan
Parking functions of length n are well known to be in correspondence with both labelled trees on n+1 vertices and factorizations of the full cycle σn=(01…n) into n transpositions. In fact, these correspondences can be refined: Kreweras equated the area enumerator of parking functions with the inversion enumerator of labelled trees, while an elegant bijection of Stanley maps the area of parking functions
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Coloring near-quadrangulations of the cylinder and the torus Eur. J. Comb. (IF 0.848) Pub Date : 2020-11-11 Zdeněk Dvořák; Jakub Pekárek
Let G be a simple connected plane graph and let C1 and C2 be cycles in G bounding distinct faces f1 and f2. For a positive integer ℓ, let r(ℓ) denote the number of integers n such that −ℓ≤n≤ℓ, n is divisible by 3, and n has the same parity as ℓ; in particular, r(4)=1. Let rf1,f2(G)=∏fr(|f|), where the product is over the faces f of G distinct from f1 and f2, and let q(G)=1+∑f:|f|≠4|f|, where the sum
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Percolating sets in bootstrap percolation on the Hamming graphs and triangular graphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-10-21 Mohammadreza Bidgoli; Ali Mohammadian; Behruz Tayfeh-Rezaie
The r-neighbor bootstrap percolation on a graph is an activation process of the vertices. The process starts with some initially activated vertices and then, in each round, any inactive vertex with at least r active neighbors becomes activated. A set of initially activated vertices leading to the activation of all vertices is said to be a percolating set. Denote the minimum size of a percolating set
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Toughness in pseudo-random graphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-10-13 Xiaofeng Gu
A d-regular graph on n vertices with the second largest absolute eigenvalue at most λ is called an (n,d,λ)-graph. The celebrated expander mixing lemma for (n,d,λ)-graphs builds a connection between graph spectrum and edge distribution. In this paper, we present some applications of the expander mixing lemma. In particular, we make progress toward the toughness conjecture of Brouwer. The toughness t(G)
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Finiteness theorems for matroid complexes with prescribed topology Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-26 Federico Castillo; José Alejandro Samper
There are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating this fact to the language of h-vectors, there are finitely many simplicial complexes of bounded dimension with h1=k for any natural number k. In this paper we study the question at the other end of the h-vector: Are there only finitely many (d−1)-dimensional simplicial complexes with hd=k for
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On-line size Ramsey number for monotone k-uniform ordered paths with uniform looseness Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-24 Xavier Pérez-Giménez; Paweł Prałat; Douglas B. West
An ordered hypergraph is a hypergraph H with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph G in H must respect the specified order on V(G). In on-line Ramsey theory, Builder iteratively presents edges that Painter must immediately color. The t-color on-line size Ramsey number R̃t(G) of an ordered hypergraph G is the minimum number of edges Builder needs to
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3-regular graphs are 2-reconstructible Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-22 Alexandr V. Kostochka; Mina Nahvi; Douglas B. West; Dara Zirlin
A graph is ℓ-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting ℓ vertices. We prove that 3-regular graphs are 2-reconstructible.
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r-hued (r+1)-coloring of planar graphs with girth at least 8 for r≥9 Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-18 Hoang La; Mickael Montassier; Alexandre Pinlou; Petru Valicov
Let r,k≥1 be two integers. An r-hued k-coloring of the vertices of a graph G=(V,E) is a proper k-coloring of the vertices, such that, for every vertex v∈V, the number of colors in its neighborhood is at least min{dG(v),r}, where dG(v) is the degree of v. We prove the existence of an r-hued (r+1)-coloring for planar graphs with girth at least 8 for r≥9. As a corollary, every planar graph with maximum
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On the size of shadow-added intersecting families Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-16 Peter Frankl
Let n≥2k−1>1, [n]={1,2,…,n}. For a family F of k-subsets of [n] let ∂F be the immediate shadow (cf. Definition 1.1) of F. Suppose that |F∩F′|≥2 for all F,F′∈F. We conjecture that |F|+|∂F|≤3n−2k−2+n−2k−3 and prove it for n=2k−1, n≥3(k−1) and also for k≤10. This problem is somewhat unusual but we exhibit deep connections to the Erdős–Ko–Rado Theorem and to the Erdős Matching Conjecture. Some related
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Lattice paths and branched continued fractions II. Multivariate Lah polynomials and Lah symmetric functions Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-15 Mathias Pétréolle, Alan D. Sokal
We introduce the generic Lah polynomials Ln,k(ϕ), which enumerate unordered forests of increasing ordered trees with a weight ϕi for each vertex with i children. We show that, if the weight sequence ϕ is Toeplitz-totally positive, then the triangular array of generic Lah polynomials is totally positive and the sequence of row-generating polynomials Ln(ϕ,y) is coefficientwise Hankel-totally positive
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Counting primitive subsets and other statistics of the divisor graph of {1,2,…,n} Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-09 Nathan McNew
Let Q(n) denote the count of the primitive subsets of the integers {1,2…,n}. We give a new proof that Q(n)=α(1+o(1))n for some constant α, which allows us to give a good error term and to improve upon the lower bound for the value of α. We also show that the method developed can be applied to many similar problems related to the divisor graph, including other questions about primitive sets, geomet
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Graphs that contain multiply transitive matchings Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-07 Alex Schaefer, Eric Swartz
Let Γ be a finite, undirected, connected, simple graph. We say that a matching M is a permutable m-matching if M contains m edges and the subgroup of Aut(Γ) that fixes the matching M setwise allows the edges of M to be permuted in any fashion. A matching M is 2-transitive if the setwise stabilizer of M in Aut(Γ) can map any ordered pair of distinct edges of M to any other ordered pair of distinct edges
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Blowup Ramsey numbers Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-07 Victor Souza
We study a generalisation of the bipartite Ramsey numbers to blowups of graphs. For a graph G, denote the t-blowup of G by G[t]. We say that G is r-Ramsey for H, and write G→rH, if every r-colouring of the edges of G has a monochromatic copy of H. We show that if G→rH, then for all t, there exists n such that G[n]→rH[t]. In fact, we provide exponential lower and upper bounds for the minimum n with
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The γ-positivity of bivariate Eulerian polynomials via the Hetyei–Reiner action Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-04 Hua Sun
The bivariate Eulerian polynomials are defined by An(p,q)=∑π∈Snpodes(π)qedes(π),where odes(π) and edes(π) are the number of descents of permutation π in odd and even positions, respectively. In this paper, by the Hetyei–Reiner action, we show that for k≥1, the bivariate Eulerian polynomials A2k+1(p,q) and (1+p)−1A2k(p,q) are γ-positive, namely, they can be expressed in terms of the basis Bn≔{(pq)i
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Signed Euler–Mahonian identities Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-03 Sen-Peng Eu; Zhicong Lin; Yuan-Hsun Lo
A relationship between signed Eulerian polynomials and the classical Eulerian polynomials on Sn was given by Désarménien and Foata in 1992, and a refined version, called signed Euler–Mahonian identity, together with a bijective proof was proposed by Wachs in the same year. By generalizing this bijection, in this paper we extend the above results to the Coxeter groups of types Bn, Dn, and the complex
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2-connected chordal graphs and line graphs are (1,5)-choosable Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-03 Tsai-Lien Wong
Suppose G is a graph and k,k′ are positive integers. A (k,k′)-list assignment is a mapping L which assigns to each vertex v a set L(v) of k real numbers, and assigns to each edge e a set L(e) of k′ real numbers. A proper L-total weighting is a mapping ϕ:V(G)∪E(G)→R such that ϕ(z)∈L(z) for z∈V∪E, and ∑e∈E(u)ϕ(e)+ϕ(u)≠∑e∈E(v)ϕ(e)+ϕ(v) for every edge uv. A graph G is called (k,k′)-choosable if for every
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On circular-perfect graphs: A survey Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-02 Arnaud Pêcher; Annegret K. Wagler
Circular-perfect graphs form a natural superclass of perfect graphs, introduced by Zhu almost 20 years ago: on the one hand due to their definition by means of a more general coloring concept, on the other hand as an important class of χ-bound graphs with the smallest non-trivial χ-binding function χ(G)≤ω(G)+1. In this paper, we survey the results about circular-perfect graphs obtained in the two last
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On the 4-color theorem for signed graphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-02 František Kardoš; Jonathan Narboni
Máčajová et al. (2016) defined the chromatic number of a signed graph which coincides for all-positive signed graphs with the chromatic number of unsigned graphs. They conjectured that in this setting, for signed planar graphs four colors are always enough, generalizing thereby The Four Color Theorem. We disprove the conjecture.
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Random hypergraphs and property B Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-02 Lech Duraj; Jakub Kozik; Dmitry Shabanov
In 1964 Erdős proved that (1+o1)eln(2)4k22k edges are sufficient to build a k-graph which is not two colorable. To this day, it is not known whether there exist such k-graphs with smaller number of edges. Erdős’ bound is consequence of the fact that a hypergraph with k2∕2 vertices and M(k)=(1+o1)eln(2)4k22k randomly chosen edges of size k is asymptotically almost surely not two colorable. Our first
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Superposition of snarks revisited Eur. J. Comb. (IF 0.848) Pub Date : 2020-09-01 Edita Máčajová; Martin Škoviera
In this paper we propose a new approach to superposition of snarks, a powerful method of constructing large cubic graphs with no 3-edge-colouring from small ones. The main idea is to use surjective mappings between graphs similar to graph homomorphisms and to control flows induced from the domain graph to the target graph via the mappings. This leads to significant strengthening of the power of the
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Homomorphisms of signed graphs: An update Eur. J. Comb. (IF 0.848) Pub Date : 2020-08-28 Reza Naserasr; Éric Sopena; Thomas Zaslavsky
A signed graph is a graph together with an assignment of signs to the edges. A closed walk in a signed graph is said to be positive (negative) if it has an even (odd) number of negative edges, counting repetition. Recognizing the signs of closed walks as one of the key structural properties of a signed graph, we define a homomorphism of a signed graph (G,σ) to a signed graph (H,π) to be a mapping of
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The last temptation of William T. Tutte Eur. J. Comb. (IF 0.848) Pub Date : 2020-08-27 Bojan Mohar; Nathan Singer
In 1999, at one of his last public lectures, Tutte discussed a question he had considered since the times of the Four Color Conjecture. He asked whether the 4-coloring complex of a planar triangulation could have two components in which all colorings had the same parity. In this note we answer Tutte’s question contrary to his speculations by showing that there are triangulations of the plane whose
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Optimal radio-k-labelings of trees Eur. J. Comb. (IF 0.848) Pub Date : 2020-08-27 Angel Chavez; Daphne Der-Fen Liu; Mason Shurman
Let G be a graph, and let k be a positive integer. The radio-k-number of G is the smallest integer s for which there exists a function f:V(G)→{0,1,2,…,s} such that for any two vertices u,v∈V(G), |f(u)−f(v)|⩾k+1−d(u,v), where d(u,v) is the distance between u and v. In particular, when d is the diameter of G, the radio-d-number is called the radio number of G. This article contains four major parts.
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On-line partitioning of width w posets into wO(loglogw) chains Eur. J. Comb. (IF 0.848) Pub Date : 2020-08-27 Bartłomiej Bosek; Tomasz Krawczyk
An on-line chain partitioning algorithm receives the elements of a poset one at a time, and when an element is received, irrevocably assigns it to one of the chains. In this paper, we present an on-line algorithm that partitions posets of width w into wO(loglogw) chains. This improves over previously best known algorithms using wO(logw) chains by Bosek and Krawczyk and by Bosek, Kierstead, Krawczyk
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Odd decompositions and coverings of graphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-08-26 Mirko Petruševski; Riste Škrekovski
A (finite) graph is odd if all its vertices have odd degrees. The principal aim of this survey is to present the current state of research on covers and decompositions of graphs into fewest possible number of odd subgraphs. Given a graph G, the parameters χo′(G) and covo(G) denote, respectively, the minimum size of a decomposition and cover of G consisting of odd subgraphs. Pyber (1991) and Mátrai
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On subgroup perfect codes in Cayley graphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-08-25 Junyang Zhang; Sanming Zhou
A perfect code in a graph Γ=(V,E) is a subset C of V such that no two vertices in C are adjacent and every vertex in V∖C is adjacent to exactly one vertex in C. A subgroup H of a group G is called a subgroup perfect code of G if there exists a Cayley graph of G which admits H as a perfect code. Equivalently, H is a subgroup perfect code of G if there exists an inverse-closed subset A of G containing
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Stable structure on safe set problems in vertex-weighted graphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-08-25 Shinya Fujita; Boram Park; Tadashi Sakuma
Let G be a graph, and let w be a positive real-valued weight function on V(G). For every subset S of V(G), let w(S)=∑v∈Sw(v). A non-empty subset S⊂V(G) is a weighted safe set of (G,w) if, for every component C of the subgraph induced by S and every component D of G−S, we have w(C)≥w(D) whenever there is an edge between C and D. If the subgraph of G induced by a weighted safe set S is connected, then
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An analogue of the Erdős–Gallai theorem for random graphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-08-25 József Balogh; Andrzej Dudek; Lina Li
Recently, variants of many classical extremal theorems have been proved in the random environment. We, complementing existing results, extend the Erdős–Gallai Theorem in random graphs. In particular, we determine, up to a constant factor, the maximum number of edges in a Pn-free subgraph of G(N,p), practically for all values of N,n and p. Our work is also motivated by the recent progress on the size-Ramsey
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Classes of graphs with low complexity: The case of classes with bounded linear rankwidth Eur. J. Comb. (IF 0.848) Pub Date : 2020-08-24 Jaroslav Nešetřil; Patrice Ossona de Mendez; Roman Rabinovich; Sebastian Siebertz
Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths – a result that shows a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on classes with bounded linear rankwidth and prove structural
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Concepts of signed graph coloring Eur. J. Comb. (IF 0.848) Pub Date : 2020-08-24 Eckhard Steffen; Alexander Vogel
This paper surveys recent development of concepts related to coloring of signed graphs. Various approaches are presented and discussed.
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On 2-defective DP-colorings of sparse graphs Eur. J. Comb. (IF 0.848) Pub Date : 2020-08-24 Alexandr Kostochka; Jingwei Xu
Introduced by Dvořák and Postle, the notion of DP-coloring generalizes list coloring and helps to prove new results on list coloring. We consider 1-defective and 2-defective DP-colorings of graphs with 2 colors. For j=1,2, we find exact lower bounds on the number of edges in (j,2)-DP-critical graphs (that is, graphs that do not admit j-defective DP-colorings with 2 colors but whose every proper subgraph
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