Showing papers in "Electronic Journal of Combinatorics in 1997"
TL;DR: In this paper, a bijection between Eulerian planar maps with prescribed vertex degrees and some plane trees that are called balanced eulerian trees is given. But the result is restricted to planar map with restricted vertex degrees.
Abstract: We give a bijection between Eulerian planar maps with prescribed vertex degrees, and some plane trees that we call balanced Eulerian trees. To enumerate the latter, we introduce conjugation classes of planted plane trees. In particular, the result answers a question of Bender and Canfield and allows uniform random generation of Eulerian planar maps with restricted vertex degrees. Using a well known correspondence between 4-regular planar maps with n vertices and planar maps with n edges we obtain an algorithm to generate uniformly such maps with complexity O(n). Our bijection is also refined to give a combinatorial interpretation of a parameterization of Arques of the generating function of planar maps with respect to vertices and faces.
131 citations
TL;DR: No position in any word can be the beginning of the rightmost occurrence of more than two squares, from which it is proved that M(n) 0, and P( n) = n-o(N) for infinitely many n.
Abstract: All our words (strings) are over a fixed alphabet. A square is a subword of the form uu = u2, where u is a nonempty word. Two squares are distinct if they are of different shape, not just translates of each other. A word u is primitive if u cannot be written in the form u = vj for some j ? 2. A square u2 with u primitive is primitive rooted. Let M(n) denote the maximum number of distinct squares, P(n) the number of distinct primitive rooted squares in a word of length n. We prove: no position in any word can be the beginning of the rightmost occurrence of more than two squares, from which we deduce M(n) 0, and P(n) = n-o(n) for infinitely many n.
83 citations
TL;DR: A polynomial algorithm is presented that computes a compact representation of the potentially exponential-sized set ${\cal C_R}$ in $O(
u m^3)$ (where $
u$ denotes the cyclomatic number).
Abstract: The perception of cyclic structures is a crucial step in the analysis of graphs. To describe the cycle vector space of a graph, a minimum cycle basis can be computed in polynomial time using an algorithm of [Horton, 1987]. But the set of cycles corresponding to a minimum basis is not always relevant for analyzing the cyclic structure of a graph. This restriction is due to the fact that a minimum cycle basis is generally not unique for a given graph. Therefore, the smallest canonical set of cycles which describes the cyclic structure of a graph is the union of all the minimum cycle bases . This set of cycles is called the set of relevant cycles and denoted by ${\cal C_R}$. A relevant cycle can also be defined as a cycle which is not the sum of shorter cycles. A polynomial algorithm is presented that computes a compact representation of the potentially exponential-sized set ${\cal C_R}$ in $O(
u m^3)$ (where $
u$ denotes the cyclomatic number). This compact representation consists of a polynomial number of relevant cycle prototypes from which all the relevant cycles can be listed in $O(n\,|{\cal C_R}|)$. A polynomial method is also given that computes the number of relevant cycles without listing all of them.
72 citations
TL;DR: It is shown that the asymptotic expression for the duration of the algorithm exhibits some periodic fluctuations and consequently no limiting distribution exists.
Abstract: We consider a leader election algorithm in which a set of distributed objects (people, computers, etc.) try to identify one object as their leader. The election process is randomized, that is, at every stage of the algorithm those objects that survived so far flip a biased coin, and those who received, say a tail, survive for the next round. The process continues until only one objects remains. Our interest is in evaluating the limiting distribution and the rst two moments of the number of rounds needed to select a leader. We establish precise asymptotics for the rst two moments, and show that the asymptotic expression for the duration of the algorithm exhibits some periodic fluctuations and consequently no limiting distribution exists. These results are proved by analytical techniques of the precise analysis of algorithms such as: analytical poissonization and depoissonization, Mellin transform, and complex analysis.
71 citations
Journal Article•
TL;DR: General upper bounds on v(k,g) are established which are roughly the 3/2 power of the lower bounds, and explicit constructions for such (k, g)-graphs are provided.
Abstract: Let k≥2 and g≥3 be integers. A (k,g)-graph is a k-regular graph with girth (length of a smallest cycle) exactly g. A (k,g)-cage is a (k,g)-graph of minimum order. Let v(k,g) be the order of a (k,g)-cage. The problem of determining v(k,g) is unsolved for most pairs (k,g) and is extremely hard in the general case. It is easy to establish the following lower bounds for v(k,g): v(k,g)≥ k(k−1) (g−1)/2−2 k−2 for g odd, and v(k,g)≥ 2(k−1) g/2−2 k−2 for g even. The best known upper bounds are roughly the squares of the lower bounds. In this paper we establish general upper bounds on v(k,g) which are roughly the 3/2 power of the lower bounds, and we provide explicit constructions for such (k,g)-graphs. Mathematical Reviews Subject Numbers: 05C35, 05C38. Secondary: 05D99.
48 citations
TL;DR: This work identifies a set of signed points, called Toeplitz points, in ${{Z}}^d, with the following property: for every $n>0$ steps, the excess of the number of lattice walks of $n$ steps is equal to ${n\choose n/2} times thenumber of permutations of $\{1,2,\dots,n\}$ that contain no ascending subsequence of length $>d$.
Abstract: We identify a set of $d!$ signed points, called Toeplitz points , in ${{Z}}^d$, with the following property: for every $n>0$, the excess of the number of lattice walks of $n$ steps, from the origin to all positive Toeplitz points, over the number to all negative Toeplitz points, is equal to ${n\choose n/2}$ times the number of permutations of $\{1,2,\dots ,n\}$ that contain no ascending subsequence of length $>d$. We prove this first by generating functions, using a determinantal theorem of Gessel. We give a second proof by direct construction of an appropriate involution. The latter provides a purely combinatorial proof of Gessel's theorem by interpreting it in terms of lattice walks. Finally we give a proof that uses the Schensted algorithm.
41 citations
TL;DR: In this paper, the authors proved a constant term conjecture of Robbins and Zeilberger (J. Com- bin. Theory Ser. A 66 (1994), 17{27) by translating the problem into a determinant evaluation problem and evaluating the determinant.
Abstract: We prove a constant term conjecture of Robbins and Zeilberger (J. Com- bin. Theory Ser. A 66 (1994), 17{27), by translating the problem into a determinant evaluation problem and evaluating the determinant. This determinant generalizes the determinant that gives the number of all totally symmetric self-complementary plane partitions contained in a (2n)(2n)(2n) box and that was used by Andrews (J. Com- bin. Theory Ser. A 66 (1994), 28{39) and Andrews and Burge (Pacic J. Math. 158 (1993), 1{14) to compute this number explicitly. The evaluation of the generalized determinant is independent of Andrews and Burge's computations, and therefore in particular constitutes a new solution to this famous enumeration problem. We also evaluate a related determinant, thus generalizing another determinant identity of An- drews and Burge (loc. cit.). By translating some of our determinant identities into constant term identities, we obtain several new constant term identities.
41 citations
TL;DR: Some basic concepts and properties related with multimatroids are investigated: matroid orthogonality, minor operations and connectivity.
Abstract: A multimatroid is a combinatorial structure that encompasses matroids, delta-matroids and isotropic systems. This structure has been introduced to unify a theorem of Edmonds on the coverings of a matroid by independent sets and a theorem of Jackson on the existence of pairwise compatible Euler tours in a 4-regular graph. Here we investigate some basic concepts and properties related with multimatroids: matroid orthogonality, minor operations and connectivity.
38 citations
TL;DR: It is shown that is an r-antipodal distanceregular graph if and only if the distance graph d is constituted by disjoint copies of the complete graph Kr, with r satisfying an expression in terms of n and the distinct eigenvalues.
Abstract: Let be a regular (connected) graph with n vertices and d + 1 distinct eigenvalues. As a main result, it is shown that is an r-antipodal distanceregular graph if and only if the distance graph d is constituted by disjoint copies of the complete graph Kr, with r satisfying an expression in terms of n and the distinct eigenvalues.
37 citations
TL;DR: The tools used in the construction are balanced generalized weighing matrices and regular Hadamard matrices of order $9\cdot 4^d$.
Abstract: For every positive integer $m$, we construct a symmetric $(v,k,\lambda )$-design with parameters $v={{h((2h-1)^{2m}-1)}\over{h-1}}$, $k=h(2h-1)^{2m-1}$, and $\lambda =h(h-1)(2h-1)^{2m-2}$, where $h=\pm 3\cdot 2^d$ and $|2h-1|$ is a prime power. For $m\geq 2$ and $d\geq 1$, these parameter values were previously undecided. The tools used in the construction are balanced generalized weighing matrices and regular Hadamard matrices of order $9\cdot 4^d$.
33 citations
TL;DR: It is shown that for any fixed $k \ge 2$ there is a constant $c_k$ such that for sufficiently large $n, $\delta _k(n) = k$ and the lower bound is obtained by use of circulant graphs.
Abstract: Let $\gamma (n,\delta)$ denote the largest possible domination number for a graph of order $n$ and minimum degree $\delta$. This paper is concerned with the behavior of the right side of the sequence $$\gamma (n,0) \ge \gamma (n,1) \ge \cdots \ge \gamma (n,n-1) = 1. $$ We set $ \delta _k(n) = \max \{ \delta \, \vert \, \gamma (n,\delta) \ge k \}$, $k \ge 1.$ Our main result is that for any fixed $k \ge 2$ there is a constant $c_k$ such that for sufficiently large $n$, $$ n-c_kn^{(k-1)/k} \le \delta _{k+1}(n) \le n - n^{(k-1)/k}. $$ The lower bound is obtained by use of circulant graphs. We also show that for $n$ sufficiently large relative to $k$, $\gamma (n,\delta _k(n)) = k$. The case $k=3$ is examined in further detail. The existence of circulant graphs with domination number greater than 2 is related to a kind of difference set in ${\bf Z}_n$.
TL;DR: The first complete combinatorial proof of the fact that the number of domino tilings of the 2n×2n square grid is of the form 2^n(2k + 1)^2 is given, thus settling a question raised by John, Sachs, and Zernitz.
Abstract: We give the first complete combinatorial proof of the fact that the number of domino tilings of the 2n×2n square grid is of the form 2^n(2k + 1)^2, thus settling a question raised by John, Sachs, and Zernitz. The proof lends itself naturally to some interesting generalizations, and leads to a number of new conjectures.
TL;DR: F (n) is conjectured to be c2 c2 n for some constant c and a construction is given that yields f(n) that yields c(n).
Abstract: A set S of positive integers has distinct subset sums if there are $2^{|S|}$ distinct elements of the set $\left\{ \sum_{x \in X} x: X \subset S \right\} . $ Let $$f(n) = \min\{ \max S: |S|=n {\rm \hskip2mm and \hskip2mm} S {\rm \hskip2mm has \hskip2mm distinct \hskip2mm subset \hskip2mm sums}\}.$$ Erdős conjectured $ f(n) \ge c2^{n}$ for some constant c. We give a construction that yields $f(n)
TL;DR: It is shown that all of the currently known largest graphs of degree $\le 15$ and diameter $\le 10$ have been found by computer search among Cayley graphs of semidirect products of cyclic groups, and all of them can in fact be described as lifts of smaller Cayley graph of cycling groups, with voltages in (other) cyclicgroups.
Abstract: Voltage graphs are a powerful tool for constructing large graphs (called lifts ) with prescribed properties as covering spaces of small base graphs. This makes them suitable for application to the degree/diameter problem , which is to determine the largest order of a graph with given degree and diameter. Many currently known largest graphs of degree $\le 15$ and diameter $\le 10$ have been found by computer search among Cayley graphs of semidirect products of cyclic groups. We show that all of them can in fact be described as lifts of smaller Cayley graphs of cyclic groups, with voltages in (other) cyclic groups. This opens up a new possible direction in the search for large vertex-transitive graphs of given degree and diameter.
TL;DR: The behaviour of the random greedy algorithm for constructing a maximal packing of edgedisjoint triangles on n points (a maximal partial triple system) is analysed with particular emphasis on the flnal number of unused edges.
Abstract: The behaviour of the random greedy algorithm for constructing a maximal packing of edgedisjoint triangles on n points (a maximal partial triple system) is analysed with particular emphasis on the flnal number of unused edges. It is shown that this number is at mostn 7=4+o(1) , \halfway" from the previous best-known upper bound o(n 2 ) to the conjectured value n 3=2+o(1) . The more general problem of random greedy packing in hypergraphs is also considered.
TL;DR: A method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice to remove initialization bias from Monte Carlo randomization is described.
Abstract: This survey article describes a method for choosing uniformly at random from any finite set whose objects can be viewed as constituting a distributive lattice. The method is based on ideas of the author and David Wilson for using "coupling from the past" to remove initialization bias from Monte Carlo randomization. The article describes several applications to specific kinds of combinatorial objects such as tilings, constrained lattice paths, an alternating sign matrices.
TL;DR: By using asymptotic methods to study the degrees of sequences of characters of symmetric groups, some unexpected arithmetic properties between these products are obtained.
Abstract: Asymptotic calculations are applied to study the degrees of cer- tain sequences of characters of symmetric groups. Starting with a given partition , we deduce several skew diagrams which are related to . To each such skew dia- gram there corresponds the product of its hook numbers. By asymptotic methods we obtain some unexpected arithmetic properties between these products. The authors do not know "nite", nonasymptotic proofs of these results. The problem appeared in the study of the hook formula for various kinds of Young diagrams. The proofs are based on properties of shifted Schur functions, due to Okounkov and Olshanski. The theory of these functions arose from the asymptotic theory of Vershik and Kerov of the representations of the symmetric groups.
TL;DR: An upper bound for the cardinality of k (u) is derived, showing that j k (U)j decreases at least as O( 2 ), and the cases in which the bound is attained are characterized.
Abstract: Given a vertex u 2 V of a graph = ( V; E), the (local) proper polynomials constitute a sequence of orthogonal polynomials, constructed from the so-called u-local spectrum of . These polynomials can be thought of as a generalization, for all graphs, of the distance polynomials for the distance-regular graphs. The (local) adjacency polynomials, which are basically sums of proper polynomials, were recently used to study a new concept of distance-regularity for non-regular graphs, and also to give bounds on some distance-related parameters such as the diameter. Here we develop the subject of these polynomials and gave a survey of some known results involving them. For instance, distance-regular graphs are characterized from its spectrum and the number of vertices at \extremal distance" from each of their vertices. Afterwards, some new applications of both, the proper and adjacency polynomials, are derived, such as bounds for the radius of and the weight k-excess of a vertex. Given the integers k; 0, let k (u) denote the set of vertices which are at distance at least k from a vertex u 2 V , and there exist exactly (shortest) k-paths from u to each of such vertices. As a main result, an upper bound for the cardinality of k (u) is derived, showing that j k (u)j decreases at least as O( 2 ), and the cases in which the bound is attained are characterized. When these results are particularized to regular graphs with four distinct eigenvalues, we reobtain a result of Van Dam about 3-class association schemes, and prove some conjectures of Haemers and Van Dam, about the number of vertices at distance three from every vertex of a regular graph with four distinct eigenvalues |setting k = 2 and = 0| and, more generally, the number of non-adjacent vertices to every vertex u 2 V , which have common neighbours with it.
TL;DR: Crapo's fl invariant is extended from matroids to greedoids, con- centrating especially on antimatroids associated with chordal graphs and combinatorial interpretations for fl(G )f or simplicial shelling antimatroid associated with chords.
Abstract: We extend Crapo's fl invariant from matroids to greedoids, con- centrating especially on antimatroids. Several familiar expansions for fl(G) have greedoid analogs. We give combinatorial interpretations for fl(G )f or simplicial shelling antimatroids associated with chordal graphs. When G is this antimatroid and b(G) is the number of blocks of the chordal graph G ,w e prove fl(G )=1 i b ( G).
TL;DR: In this paper, the color-critical graph is defined as a graph whose vertices are more than 2k vertices and whose color is at most 2k-color-critical.
Abstract: A graph $G$ is $k$-color-critical (or simply $k$-critical ) if $\chi(G)=k$ but $\chi(G') Theorem Suppose $k\geq 4$, and let $G=(V,E)$ be a $k$-critical graph on more than $k$ vertices. Then $ |E(G)|\geq ({{k-1}\over {2}}+{{k-3}\over {2(k^2-2k-1)}})|V(G)| $
TL;DR: It is proved that the asymptotic probability that there are no tt submatrices with all ones is zero or one, according as z (t=ne) 2=t expfan=t 2 g or z (n;t) 2 =t expf(logt bn)=t 2g, where an tends to innity at a specied rate, and bn!1 is arbitrary.
Abstract: Let G2(n) denote a bipartite graph with n vertices in each color class, and let z(n;t) be the bipartite Tur an number, representing the maximum possible number of edges in G2(n) if it does not contain a copy of the complete bipartite subgraph K(t;t). It is then clear that (n;t )= n 2 z (n;t) denotes the minimum number of zeros in an nn zero-one matrix that does not contain a tt submatrix consisting of all ones. We are interested in the behaviour of z(n;t) when both t and n go to innity. The case 2 t n 1=5 has been treated in [9] ; here we use a dierent method to consider the overlapping case logn t n 1=3 . Fill an nn matrix randomly with z ones and = n 2 z zeros. Then, we prove that the asymptotic probability that there are no tt submatrices with all ones is zero or one, according as z (t=ne) 2=t expfan=t 2 g or z (t=ne) 2=t expf(logt bn)=t 2 g, where an tends to innity at a specied rate, and bn!1is arbitrary. The proof employs
TL;DR: A simple inductive proof of the Regev and Vershik conjecture about the set of hook lengths for certain skew Young diagrams, using asymptotic calculations of character degrees is given.
Abstract: Regev and Vershik (Electronic J. Combinatorics 4 (1997), #R22) have obtained some properties of the set of hook lengths for certain skew Young diagrams, using asymptotic calculations of character degrees. They also conjectured a stronger form of one of their results. We give a simple inductive proof of this conjecture. Very recently, Regev and Zeilberger (Annals of Combinatorics, to appear) have independently proved this conjecture.
TL;DR: This work reformulates the solution approach using matrix dynamics to extend results to a characterization of the asymptotic behaviour of all solutions of the Golomb recursion, and constructs the first example of a "nontrivial" nonincreasing solution, that is, one that is not eventually increasing.
Abstract: In an unpublished note Golomb proposed a family of "strange" recursions of metafibonacci type, parametrized by $k$. Previously we showed that contrary to Golomb's conjecture, for each $k$ there are many increasing solutions, and an explicit construction for multiple solutions was displayed. By reformulating our solution approach using matrix dynamics, we extend these results to a characterization of the asymptotic behaviour of all solutions of the Golomb recursion. This matrix dynamics perspective is also used to construct what we believe is the first example of a "nontrivial" nonincreasing solution, that is, one that is not eventually increasing.
TL;DR: In this article, the authors give a bound for equidistant code pairs in terms of the eigenvalues of a matrix associated with a vertex set V of a graph G, where V is the set of words of length n over an alphabet.
Abstract: Two disjoint subsets A and B of a vertex set V of a flnite graph G are called disconnected if there is no edge between A and B .I f V is the set of words of length n over an alphabetf1;:::;qgand if two words are adjacent whenever their Hamming distance is not equal to afl xed ‐2f 1;:::;ng, then a pair of disconnected sets becomes an equidistant code pair. For disconnected sets A and B we will give a bound forjAj¢j Bj in terms of the eigenvalues of a matrix associated with G. In case the complement of G is given by a relation of an association scheme the bound takes an easy form, which applied to the Hamming scheme leads to a bound for equidistant code pairs. The bound turns out to be sharp for some values of q, n and ‐ ,a nd for q!1for any flxed n and ‐. In addition, our bound reproves some old results of Ahlswede and others, such as the maximal value ofjAj¢jBjfor equidistant code pairs A ans B in the binary Hamming Scheme.
TL;DR: The class of "recursively constructed" merging networks are studied and it is shown that at least $3^{n/2-1}$ of these nets are log-periodic sorters; that is, they sort any input sequence after $n$ passes through the net.
Abstract: Let $\alpha$ and $\beta$ be a partition of $\{1,\ldots,n\}$ into two blocks. A merging network is a network of comparators which allows as input arbitrary real numbers and has the property that, whenever the input sequence $x_1,x_2,\ldots,x_n$ is such that the subsequence in the positions $\alpha$ and the subsequence in the positions $\beta$ are each sorted, the output sequence will be sorted. We study the class of "recursively constructed" merging networks and characterize those with delay $\lceil\log_2 n\rceil$ (the best possible delay for all merging networks). When $n$ is a power of 2, we show that at least $3^{n/2-1}$ of these nets are log-periodic sorters; that is, they sort any input sequence after $\log_2n$ passes through the net. (Two of these have appeared previously in the literature.)
TL;DR: In this paper, the problem of characterizing inversion multigraphs associated to affine permutations associated to Affine permutation is solved.
Abstract: In this paper we solve the problem of characterizing inversion multigraphs associated to affine permutations.
TL;DR: This paper proves a conjecture that with respect to a rather natural operation, the set of $N$-dimensional magic cubes forms a free monoid for every integer $N>1$.
Abstract: In this paper we prove a conjecture stated in an earlier paper [A-L]. The conjecture states that with respect to a rather natural operation, the set of $N$-dimensional magic cubes forms a free monoid for every integer $N>1$. A consequence of this conjecture is a certain identity of formal Dirichlet series. These series and the associated power series are shown to diverge. Generalizations of the underlying ideas are presented. We also prove variants of the main results for magic cubes with remarkable power sum properties.
TL;DR: It is shown that there is an absolute constant $\epsilon >0$ so that any tournament on $n$ vertices contains a score certificate with at most $ ({1/2}-\ep silon)n^2$ edges.
Abstract: An isomorphism certificate of a labeled tournament $T$ is a labeled subdigraph of $T$ which together with an unlabeled copy of $T$ allows the errorless reconstruction of $T$. It is shown that any tournament on $n$ vertices contains an isomorphism certificate with at most $n \log_2 n$ edges. This answers a question of Fishburn, Kim and Tetali. A score certificate of $T$ is a labeled subdigraph of $T$ which together with the score sequence of $T$ allows its errorless reconstruction. It is shown that there is an absolute constant $\epsilon >0$ so that any tournament on $n$ vertices contains a score certificate with at most $ ({1/2}-\epsilon)n^2$ edges.
TL;DR: Two modular versions of Ray-Chaudhuri-Wilson inequality for polynomial semi-lattices are proved, both of which are extensions of the original one.
Abstract: Let $X$ be an $n$-set and $L$ a set of nonnegative integers. ${\cal F}$, a set of subsets of $X$, is said to be an $L$ -intersection family if and only if for all $E
eq F \in {\cal F}, \, |E \cap F | \in L$. A special case of a conjecture of Frankl and Furedi states that if $ L = \{1, 2, \dots,k\}$,$ k$ a positive integer, then $|{\cal F}| \leq\sum_{i=0}^{k}{n-1\choose i}$. Here $|{\cal F}|$ denotes the number of elements in ${\cal F}$. Recently Ramanan proved this conjecture. We extend his method to polynomial semi-lattices and we also study some special $L$-intersection families on polynomial semi-lattices. Finally we prove two modular versions of Ray-Chaudhuri-Wilson inequality for polynomial semi-lattices.
TL;DR: An exact lower bound is proved on $\gamma(G)$, the size of the smallest matching that a certain $O(m+n)$ time greedy matching procedure may find for a given graph G, that is, over all graphs with n vertices and m edges.
Abstract: We prove an exact lower bound on $\gamma(G)$, the size of the smallest matching that a certain $O(m+n)$ time greedy matching procedure may find for a given graph $G$ with $n$ vertices and $m$ edges. The bound is precisely Erdős and Gallai's extremal function that gives the size of the smallest maximum matching, over all graphs with $n$ vertices and $m$ edges. Thus the greedy procedure is optimal in the sense that when only $n$ and $m$ are specified, no algorithm can be guaranteed to find a larger matching than the greedy procedure. The greedy procedure and augmenting path algorithms are seen to be complementary: the greedy procedure finds a large matching for dense graphs, while augmenting path algorithms are fast for sparse graphs. Well known hybrid algorithms consisting of the greedy procedure followed by an augmenting path algorithm are shown to be faster than the augmenting path algorithm alone. The lower bound on $\gamma(G)$ is a stronger version of Erdős and Gallai's result, and so the proof of the lower bound is a new way of proving of Erdős and Gallai's result.