# Discrete Mathematics

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Hypergraph Coloring Complexes
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August 2012

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Felix Breuer

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The aim of this paper is to generalize the notion of the coloring complex of a graph to hypergraphs. We present three different interpretations of those complexes-a purely combinatorial one and two geometric ones. It is shown, that most of the properties, which are known to be true for coloring complexes of graphs, break down in this more general setting, e.g., Cohen-Macaulayness and partitionability. Nevertheless, we are able to provide bounds for the [Formula: see text]- and [Formula: see text]-vectors of those complexes which yield new bounds on chromatic polynomials of hypergraphs. Moreover, though it is proven that the coloring complex of a hypergraph has a wedge decomposition, we provide an example showing that in general this decomposition is not homotopy equivalent to a wedge of spheres. In addition, we can completely characterize those hypergraphs whose coloring complex is connected.

On the -complete hypergroups

October 1999

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The class of -complete hypergroups is introduced. Several properties and examples are found and a geometric interpretation is given by means of hypergraphs.

Multivariate linear recurrences and power series division

December 2012

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M. Bousquet-Mélou and M. Petkovšek [Discrete Math. 225, No. 1–3, 51–75 (2000; Zbl 0963.05005)] investigated the generating functions of multivariate linear recurrences with constant coefficients. We will give a reinterpretation of their results by means of division theorems for formal power series, which clarifies the structural background and provides short, conceptual proofs. In addition, extending the division to the context of differential operators, the case of recurrences with polynomial coefficients can be treated in an analogous way.

Rectangular groupoids and related structures

July 2013

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The quasivariety of groupoids (N,∗) satisfying the implication a∗b=c∗d⇒a∗d=c∗b=a∗b generalises rectangular semigroups and central groupoids. We call them rectangular groupoids and find three combinatorial structures based upon arrays, matrices and graphs that are closely related. These generalise several groupoids of independent interest. The quasivariety generates the variety of all groupoids; they satisfy no nontrivial equations. We see some strong connections with isotopy, this being one of the classes of algebras (along with quasigroups) closed under isotopy. We investigate some constructions and show that a regular automorphism exists iff the groupoid is derived from a group via a Cayley graph construction.

On 2-factors with k components

May 2008

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We study the minimum degree condition for a Hamiltonian graph to have a 2-factor with k components. By proving a conjecture of R. J. Faudree, R. J. Gould, M. S. Jacobson, L. Lesniak and A. Saito [A note on 2-factors with two components, Discrete Math. 300, No. 1–3, 218–224 (2005; Zbl 1081.05062)] we show the following. There exists a real number ε>0 such that for every integer k⩾2 there exists an integer n 0 =n 0 (k) such that every Hamiltonian graph G of order n⩾n 0 with δ(G)⩾(1 2-ε has a 2-factor with k components.

Point determining digraphs, $\{0,1\}$-matrix partitions, and dualities in full homomorphisms

August 2013

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A digraph is point determining if for any two distinct vertices there exists a vertex which has an arc to (or from) exactly one of . We prove that every point-determining digraph contains a vertex such that is also point determining. We apply this result to show that for any -matrix , with diagonal zeros and diagonal ones, the size of a minimal -obstruction is at most . This is a best possible bound, and it extends the results of Sumner, and of Feder and Hell, from undirected graphs and symmetric matrices to digraphs and general matrices.

A reduced formula for the precise number of (0,1)-matrices in A(R,S)

September 2002

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A formula that calculates the number of n x m matrices in A(R, S) was presented by Wang (Sci. sinica Ser. A 1 (1988) 1). This formula has 2(n) - 2n variables. Later, in 1998, a reduced formula was proposed by Wang and Zhang, in which the number of involved variables was reduced to only 2(n-1) -n. The reduction in the number of variables is important, but it continues being of order 2(n). In this paper a new reduced formula is presented. This formula contains only (n - 2)(n - 1)/2 variables, that is, of order n(2)

Universal tilings and universal (0,1)-matrices

January 1986

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A periodic regular tiling of the plane by black and white squares is k-universal if it contains all possible k × k blocks of black and white tiles. There is a 4 × 4 periodic tiling that is 2-universal; this paper looks for the smallest 3-universal tiling and obtains a 64 × 32 periodic tiling that is 3-universal.Related to this is the following: a (0,1)-matrix is k-universal if every possible k × k (0,1)-matrix occurs as a submatrix. It is proved that, for k even, there is a k2k/2 matrix that is k-universal and, for k odd, there is a (3k + 1)2(k−3)/2 by (3k − 1)2k−3/2 matrix that is k-universal.

Planar graphs without 5-cycles and intersecting triangles are $(1,1,0)$-colorable

September 2014

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A $(c_1,c_2,...,c_k)$-coloring of $G$ is a mapping $\varphi:V(G)\mapsto\{1,2,...,k\}$ such that for every $i,1 \leq i \leq k$, $G[V_i]$ has maximum degree at most $c_i$, where $G[V_i]$ denotes the subgraph induced by the vertices colored $i$. Borodin and Raspaud conjecture that every planar graph without $5$-cycles and intersecting triangles is $(0,0,0)$-colorable. We prove in this paper that such graphs are $(1,1,0)$-colorable.

The existence of restricted resolvable designs I: (1,2)-factorizations of K2n

April 1990

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It has been known for at least a century that (in modern terminology) the complete graph K2n admits a 1-factorization, that is, a partition of its edge set E into subsets E1,…, E2n−1 such that each Ei consists of n vertex-disjoint edges. A considerably newer result (due to Ray-Chaudhuri and Wilson) states that if n is an odd integer then the complete graph K3n admits what we will call a 2-factorization, that being a pair (T, P) where T is a decomposition of K3n into triangles (K3's) and P is a partition of T into subsets so that each Ti consists of n vertex-disjoint triangles. Between these two extremes we define a (1, 2)- factorization of Kn with cardinality k to be a pair (T, P) where T is a decomposition of Kn into edges and triangles (K2's and K3's) and P is a partition of T into subsets T1,…, Tk such that each Ti is a vertex partition of Kn. This is the first in a series of two papers in which we investigate the following question: for which integers n > 0 and ⌊n/2⌋ ⩽ k ⩽ n − 1 does the complete graph Kn admit a (1, 2)-factorization of cardinality k? We prove here that when n is even the ‘obvious’ necessary conditions for the existence of these designs are sufficient, with exactly two exceptions: n = 6, k = 3; and n = 12, k = 6.

Classification of simple 2-(11,3,3) designs

February 2009

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We present an orderly algorithm for classifying triple systems. Subsequently, we show that there exist exactly 7038,699,746 nonisomorphic simple 2-(11,3,3) designs. The method is also used to confirm the previously accomplished classifications of 2-(8,3,6), 2-(12,3,2) and 2-(19,3,1) designs.

Yeh, R.K.: A survey on labeling graphs with a condition at distance two. Discrete Math. 306, 1217-1231

June 2006

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For positive integers k, d1, d2, a k- L ( d1, d2 )-labeling of a graph G is a function f : V ( G ) → { 0, 1, 2, ..., k } such that | f ( u ) - f ( v ) | {greater than or slanted equal to} di whenever the distance between u and v is i in G, for i = 1, 2. The L ( d1, d2 )-number of G, λd1 ,d2 ( G ), is the smallest k such that there exists a k- L ( d1, d2 )-labeling of G. This class of labelings is motivated by the code (or frequency) assignment problem in computer network. This article surveys the results on this labeling problem.

An aperiodic set of 13 Wang tiles

November 1996

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A new aperiodic tile set containing only 13 tiles over 5 colors is presented. Its construction is based on a recent technique developed by Kari. The tilings simulate the behavior of sequential machines that multiply real numbers in balanced representations by real constants.

Permutations with one or two 132-subsequences. Discrete Math 181(1-3):267-274

February 1998

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We prove a strikingly simple formula for the number of permutations containing exactly one subsequence of type 132. We show that this number equals the number of partitions of a convex (n + 1)-gon into n − 2 parts by noncrossing diagonals. We also prove a recursive formula for the number dn of those containing exactly two such subsequences, yielding that {dn} is P-recursive.

The quaternary complex Hadamard matrices of orders 10, 12, and 14

April 2012

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A complete classification of quaternary complex Hadamard matrices of orders 10, 12 and 14 is given, and a new parametrization scheme for obtaining new examples of affine parametric families of complex Hadamard matrices is provided. On the one hand, it is proven that all 10x10 and 12x12 quaternary complex Hadamard matrices belong to some parametric family, but on the other hand, it is shown by exhibiting an isolated 14x14 matrix that there cannot be a general method for introducing parameters into these types of matrices.

On primes dividing the group order of a doubly-even (72, 36, 16) code and the group of a quaternary (24, 12, 10) code

December 1982

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This paper finds all the odd primes p which can divide the order of the group of an extremal doubly-even (72, 36, 16) code (if one exists) and an extremal quaternary (24, 12, 10) code (if one exists). Information is given about the cycle structure of the element of order p which could aid in the construction of these codes. A number of new techniques are given for determining if an element of odd prime order can be in the group of a code.

Graphs with maximum degree Δ ≥ 17 and maximum average degree less than 3 are list 2 -distance (Δ + 2) -colorable

January 2013

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For graphs of bounded maximum average degree, we consider the problem of 2-distance coloring. This is the problem of coloring the vertices while ensuring that two vertices that are adjacent or have a common neighbor receive different colors. It is already known that planar graphs of girth at least 6 and of maximum degree D are list 2-distance (D+2)-colorable when D>=24 (Borodin and Ivanova (2009)) and 2-distance (D+2)-colorable when D>=18 (Borodin and Ivanova (2009)). We prove here that D>=17 suffices in both cases. More generally, we show that graphs with maximum average degree less than 3 and D>=17 are list 2-distance (D+2)-colorable. The proof can be transposed to list injective (D+1)-coloring.

Euler, R.: Coloring planar Toeplitz graphs and the stable set polytope. Discret. Math. 276, 183-200

February 2004

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Cliques and odd cycles are well known to induce facet-defining inequalities for the stable set polytope. In graph coloring cliques are a class of n-critical graphs whereas odd cycles represent the class of 3-critical graphs. In the first part of this paper we generalize both notions to (Kn⧹e)-cycles, a new class of n-critical graphs, and discuss some implications for the class of infinite planar Toeplitz graphs. More precisely, we show that any infinite Toeplitz graph decomposes into a finite number of connected and isomorphic components. Similar to the bipartite case, infinite planar Toeplitz graphs can be characterized by a simple condition on their defining 0–1 sequence. We then address the problem of coloring such graphs. Whereas they can always be 4-colored by a greedy-like algorithm, we are able to fully characterize the subclass of 3-chromatic such graphs. As a corollary, we obtain a König-type characterization of this class by means of (K4⧹e)-cycles. In the second part, we turn to polyhedral theory and show that (Kn⧹e)-cycles give rise to a new class of facet-defining inequalities for the stable set polytope. Then we show how Hajós’ construction can be used to further generalize (Kn⧹e)-cycles thereby providing a very large class of n-critical graphs which are still facet-inducing for the associated stable set polytope.

Otakar Bor??vka on minimum spanning tree problem Translation of both the 1926 papers, comments, history

April 2001

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Borůvka presented in 1926 the first solution of the Minimum Spanning Tree Problem (MST) which is generally regarded as a cornerstone of Combinatorial Optimization. In this paper we present the first English translation of both of his pioneering works. This is followed by the survey of development related to the MST problem and by remarks and historical perspective. Out of many available algorithms to solve MST the Borůvka's algorithm is the basis of the fastest known algorithms.

Graph factors and factorization: 1985–2003: A survey

April 2007

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In the most general sense, a factor of a graph G is just a spanning subgraph of G and a graph factorization of G is a partition of the edges of G into factors. However, as we shall see in the present paper, even this extremely general definition does not capture all the factor and factorization problems that have been studied in graph theory. Several previous survey papers have been written on this subject [F. Chung, R. Graham, Recent results in graph decompositions, London Mathematical Society, Lecture Note Series, vol. 52, Cambridge University Press, 1981, pp. 103–123; J. Akiyama, M. Kano, Factors and factorizations of graphs—a survey, J. Graph Theory 9 (1985) 1–42; L. Volkmann, Regular graphs, regular factors, and the impact of Petersen's theorems, Jahresber. Deutsch. Math.-Verein. 97 (1995) 19–42] as well as an entire book on graph decompositions [J. Bosák, Decompositions of Graphs, Kluwer Academic Publishers Group, Dordrecht, 1990]. Our purpose here is to concentrate primarily on surveying the developments of the last 15–20 years in this exponentially growing field.

On L(2,1)-labelings of Cartesian products of paths and cycles

June 2004

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A k-L(2,1)-labeling of a graph G is a function f from the vertex set V(G) to {0,1,…,k} such that |f(u)−f(v)|⩾1 if d(u,v)=2 and |f(u)−f(v)|⩾2 if d(u,v)=1. The L(2,1)-labeling problem is to find the L(2,1)-labeling number λ(G) of a graph G which is the minimum cardinality k such that G has a k-L(2,1)-labeling. In this paper, we study L(2,1)-labeling numbers of Cartesian products of paths and cycles.

Complete arcs in PG(2,25): The spectrum of the sizes and the classification of the smallest complete arcs

March 2007

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