Showing papers in "Electronic Journal of Combinatorics in 2001"
TL;DR: Luby and Vigoda proved that the Glauber dynamics, which only changes the configuration at a randomly chosen vertex in each step, has mixing time $O(n\log{n})$ when $\lambda$ is positive.
Abstract: This note considers the problem of sampling from the set of weighted independent sets of a graph with maximum degree $\Delta$. For a positive fugacity $\lambda$, the weight of an independent set $\sigma$ is $\lambda^{|\sigma|}$. Luby and Vigoda proved that the Glauber dynamics, which only changes the configuration at a randomly chosen vertex in each step, has mixing time $O(n\log{n})$ when $\lambda
97 citations
TL;DR: It is shown that the domination number of a random graph enjoys as sharp a concentration as does its chromatic number, and this fact is proved using the second moment method together with the Borel Cantelli lemma.
Abstract: In this paper, we show that the domination number $D$ of a random graph enjoys as sharp a concentration as does its chromatic number $\chi$. We first prove this fact for the sequence of graphs $\{G(n,p_n\},\; n\to\infty$, where a two point concentration is obtained with high probability for $p_n=p$ (fixed) or for a sequence $p_n$ that approaches zero sufficiently slowly. We then consider the infinite graph $G({\bf Z}^+, p)$, where $p$ is fixed, and prove a three point concentration for the domination number with probability one. The main results are proved using the second moment method together with the Borel Cantelli lemma.
70 citations
TL;DR: This work gives general lower and upper bounds on the best possible density of r-identifying codes in three infinite regular graphs.
Abstract: Consider a connected undirected graph $G=(V,E)$ and a subset of vertices $C$. If for all vertices $v \in V$, the sets $B_r(v) \cap C$ are all nonempty and pairwise distinct, where $B_r(v)$ denotes the set of all points within distance $r$ from $v$, then we call $C$ an $r$-identifying code. We give general lower and upper bounds on the best possible density of $r$-identifying codes in three infinite regular graphs.
60 citations
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TL;DR: This paper proves the $k=5$ case of the following conjecture (the lonely runner conjecture): for any $k$ positive reals $ v_1, \dots, v_k $ there exists a real number $t$ such that 1/(k+1) \le v_it \le k/(k-1) for $ i= 1, 𝕂, k$.
Abstract: For $x$ real, let $ \{x\}$ be the fractional part of $x$ (i.e. $ \{x\} = x - \lfloor x \rfloor $). In this paper we prove the $k=5$ case of the following conjecture (the lonely runner conjecture): for any $k$ positive reals $ v_1, \dots , v_k $ there exists a real number $t$ such that $ 1/(k+1) \le \{v_it \} \le k/(k+1) $ for $ i= 1, \dots, k$.
54 citations
TL;DR: It can be shown that determinants of Hankel matrices consisting of numbers ${1}\over {3m+1}} {{3m-1} \choose m} yield an alternate expression of two Mills – Robbins – Rumsey determinants important in the enumeration of plane partitions and alternating sign matrices.
Abstract: Hankel matrices consisting of Catalan numbers have been analyzed by various authors. Desainte-Catherine and Viennot found their determinant to be $\prod_{1 \leq i \leq j \leq k} {{i+j+2n}\over {i+j}}$ and related them to the Bender - Knuth conjecture. The similar determinant formula $\prod_{1 \leq i \leq j \leq k} {{i+j-1+2n}\over {i+j-1}}$ can be shown to hold for Hankel matrices whose entries are successive middle binomial coefficients ${{2m+1} \choose m}$. Generalizing the Catalan numbers in a different direction, it can be shown that determinants of Hankel matrices consisting of numbers ${{1}\over {3m+1}} {{3m+1} \choose m}$ yield an alternate expression of two Mills – Robbins – Rumsey determinants important in the enumeration of plane partitions and alternating sign matrices. Hankel matrices with determinant 1 were studied by Aigner in the definition of Catalan – like numbers. The well - known relation of Hankel matrices to orthogonal polynomials further yields a combinatorial application of the famous Berlekamp – Massey algorithm in Coding Theory, which can be applied in order to calculate the coefficients in the three – term recurrence of the family of orthogonal polynomials related to the sequence of Hankel matrices.
52 citations
TL;DR: The domino Schensted algorithm of Barbasch, Vogan, Garfinkle and van Leeuwen is described and it follows easily from this description that total color of a colored word maps to the sum of the spins of a pair of $2$-ribbon tableaux.
Abstract: We describe the domino Schensted algorithm of Barbasch, Vogan, Garfinkle and van Leeuwen. We place this algorithm in the context of Haiman's mixed and left-right insertion algorithms and extend it to colored words. It follows easily from this description that total color of a colored word maps to the sum of the spins of a pair of $2$-ribbon tableaux. Various other properties of this algorithm are described, including an alternative version of the Littlewood-Richardson bijection which yields the $q$-Littlewood-Richardson coefficients of Carre and Leclerc. The case where the ribbon tableau decomposes into a pair of rectangles is worked out in detail. This case is central in recent work by D. White on the number of even and odd linear extensions of a product of two chains.
50 citations
TL;DR: This paper provides tight bounds for the moments of the width of rooted labeled trees with $n$ nodes, answering an open question of Odlyzko and Wilf (1987).
Abstract: This paper provides tight bounds for the moments of the width of rooted labeled trees with $n$ nodes, answering an open question of Odlyzko and Wilf (1987). To this aim, we use one of the many one-to-one correspondences between trees and parking functions, and also a precise coupling between parking functions and the empirical processes of mathematical statistics. Our result turns out to be a consequence of the strong convergence of empirical processes to the Brownian bridge (Komlos, Major and Tusnady, 1975).
49 citations
TL;DR: This paper studies the action of Singer groups of projective geometries (and their subgroups) on $(d-1)-flats for arbitrary $d$ and determines the maximum co-dimension of f_q(n, h) of a flat of $PG(n-1, q)$ whose orbit under a subgroup of index $h$ of some Singer group covers all points of £PG (n- 1, q).
Abstract: We study the action of Singer groups of projective geometries (and their subgroups) on $(d-1)$-flats for arbitrary $d$. The possibilities which can occur are determined, and a formula for the number of orbits of each possible size is given. Motivated by an old problem of J.R. Isbell on the existence of certain permutation groups we pose the problem of determining, for given $q$ and $h$, the maximum co-dimension $f_q(n, h)$ of a flat of $PG(n-1, q)$ whose orbit under a subgroup of index $h$ of some Singer group covers all points of $PG(n-1, q)$. It is clear that $f_q (n, h) < n - \log_q (h)$; on the other hand we show that $f_q(n, h) \geq n - 1 - 2 \log _q (h)$.
48 citations
TL;DR: An explicit formula is derived for the polynomial part of p_A, the number of nonnegative integer solutions to positive integers, which is derived from the set of positive integers A.
Abstract: Given a set of positive integers $ A = \{ a_{1} , \dots , a_{n} \} $, we study the number $ p_{A} (t) $ of nonnegative integer solutions $ \left( m_{1} , \dots , m_{n} \right) $ to $ \sum_{j=1}^{n} m_{j} a_{j} = t $. We derive an explicit formula for the polynomial part of $p_A$.
46 citations
TL;DR: The lattice of noncrossing partitions can be embedded into the Cayley graph of the symmetric group and this allows us to rederive connections between non crossing partitions and parking functions.
Abstract: The lattice of noncrossing partitions can be embedded into the Cayley graph of the symmetric group. This allows us to rederive connections between noncrossing partitions and parking functions. We use an analogous embedding for type B non-crossing partitions in order to answer a question raised by R. Stanley on the edge labeling of the type B non-crossing partitions lattice.
46 citations
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TL;DR: It is shown that every graph with density $d$ contains a large, relatively dense $\epsilon$-regular pair, for which the bound is, up to a constant, best possible.
Abstract: The celebrated Regularity Lemma of Szemeredi asserts that every sufficiently large graph $G$ can be partitioned in such a way that most pairs of the partition sets span $\epsilon$-regular subgraphs. In applications, however, the graph $G$ has to be dense and the partition sets are typically very small. If only one $\epsilon$-regular pair is needed, a much bigger one can be found, even if the original graph is sparse. In this paper we show that every graph with density $d$ contains a large, relatively dense $\epsilon$-regular pair. We mainly focus on a related concept of an $(\epsilon,\sigma)$-dense pair, for which our bound is, up to a constant, best possible.
TL;DR: This paper proves that for $k \geq 2$, planar graphs of odd girth at least $8k-3$ have circular chromatic number at most $2+{{1}\over {k}}$.
Abstract: It was conjectured by Jaeger that $4k$-edge connected graphs admit a $(2k+1, k)$-flow. The restriction of this conjecture to planar graphs is equivalent to the statement that planar graphs of girth at least $4k$ have circular chromatic number at most $2+ {{1}\over {k}}$. Even this restricted version of Jaeger's conjecture is largely open. The $k=1$ case is the well-known Grotzsch 3-colour theorem. This paper proves that for $k \geq 2$, planar graphs of odd girth at least $8k-3$ have circular chromatic number at most $2+{{1}\over {k}}$.
TL;DR: This paper gives a combinatorial construction of the irreducible representations of the rook monoid, namely, the Specht modules.
Abstract: The wealth of beautiful combinatorics that arise in the representation theory of the symmetric group is well-known. In this paper, we analyze the representations of a related algebraic structure called the rook monoid from a combinatorial angle. In particular, we give a combinatorial construction of the irreducible representations of the rook monoid. Since the rook monoid contains the symmetric group, it is perhaps not surprising that the construction outlined in this paper is very similar to the classic combinatorial construction of the irreducible $S_n$-representations: namely, the Specht modules.
TL;DR: This short note slightly improves the best known upper bound on the Ramsey number r(C_l, K_n) for even $l$, such that every graph of order $m$ contains either cycle of length l or a set of independent vertices.
Abstract: The Ramsey number $r(C_l, K_n)$ is the smallest positive integer $m$ such that every graph of order $m$ contains either cycle of length $l$ or a set of $n$ independent vertices. In this short note we slightly improve the best known upper bound on $r(C_l, K_n)$ for odd $l$.
TL;DR: A bijection between Dyck paths and $312$-avoiding permutations is defined which takes the area statistic $a(\pi)$ on Dyckpaths to the inversion statistic on $312-avoided permutations.
Abstract: The symmetric $q,t$-Catalan polynomial $C_n(q,t)$, which specializes to the Catalan polynomial $C_n(q)$ when $t=1$, was defined by Garsia and Haiman in 1994. In 2000, Garsia and Haglund described statistics $a(\pi)$ and $b(\pi)$ on Dyck paths such that $C_n(q,t) = \sum_{\pi} q^{a(\pi)}t^{b(\pi)}$ where the sum is over all $n \times n$ Dyck paths. Specializing $t=1$ gives the Catalan polynomial $C_n(q)$ defined by Carlitz and Riordan and further studied by Carlitz. Specializing both $t=1$ and $q=1$ gives the usual Catalan number $C_n$. The Catalan number $C_n$ is known to count the number of $n \times n$ Dyck paths and the number of $312$-avoiding permutations in $S_n$, as well as at least 64 other combinatorial objects. In this paper, we define a bijection between Dyck paths and $312$-avoiding permutations which takes the area statistic $a(\pi)$ on Dyck paths to the inversion statistic on $312$-avoiding permutations. The inversion statistic can be thought of as the number of $(21)$ patterns in a permutation $\sigma$. We give a characterization for the number of $(321)$, $(4321)$, $\dots$, $(k\cdots21)$ patterns that occur in $\sigma$ in terms of the corresponding Dyck path.
TL;DR: It is shown that classical results on the nature of zeros of real orthogonal polynomials may be extended to polynomial sequences satisfying certain 4-term recursions, and specific polynometric sequences satisfying higher order recursions that should also satisfy this classical result are identified.
Abstract: This paper begins with a brief discussion of a class of polynomial Riemann hypotheses, which leads to the consideration of sequences of orthogonal polynomials and 3-term recursions. The discussion further leads to higher order polynomial recursions, including 4-term recursions where orthogonality is lost. Nevertheless, we show that classical results on the nature of zeros of real orthogonal polynomials (i. e., that the zeros of $p_n$ are real and those of $p_{n+1}$ interleave those of $p_n$) may be extended to polynomial sequences satisfying certain 4-term recursions. We identify specific polynomial sequences satisfying higher order recursions that should also satisfy this classical result. As with the 3-term recursions, the 4-term recursions give rise naturally to a linear functional. In the case of 3-term recursions the zeros fall nicely into place when it is known that the functional is positive, but in the case of our 4-term recursions, we show that the functional can be positive even when there are non-real zeros among some of the polynomials. It is interesting, however, that for our 4-term recursions positivity is guaranteed when a certain real parameter $C$ satisfies $C\ge 3$, and this is exactly the condition of our result that guarantees the zeros have the aforementioned interleaving property. We conjecture the condition $C\ge 3$ is also necessary. Next we used a classical determinant criterion to find exactly when the associated linear functional is positive, and we found that the Hankel determinants $\Delta_n$ formed from the sequence of moments of the functional when $C = 3$ give rise to the initial values of the integer sequence $1, 3, 26, 646, 45885, \cdots,$ of Alternating Sign Matrices (ASMs) with vertical symmetry. This spurred an intense interest in these moments, and we give 9 diverse characterizations of this sequence of moments. We then specify these Hankel determinants as Macdonald-type integrals. We also provide an an infinite class of integer sequences, each sequence of which gives the Hankel determinants $\Delta_n$ of the moments. Finally we show that certain $n$-tuples of non-intersecting lattice paths are evaluated by a related class of special Hankel determinants. This class includes the $\Delta_n$. At the same time, ASMs with vertical symmetry can readily be identified with certain $n$-tuples of osculating paths. These two lattice path models appear as a natural bridge from the ASMs with vertical symmetry to Hankel determinants.
TL;DR: The queen’s graph Qn has the squares of the n n chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal and each is either known, or known to have one of two values.
Abstract: The queen’s graph Qn has the squares of the n n chessboard as its vertices; two squares are adjacent if they are in the same row, column, or diagonal. Let (Qn )a ndi(Qn) be the minimum sizes of a dominating set and an independent dominating set of Qn, respectively. Recent results, the Parallelogram Law, and a search algorithm adapted from Knuth are used to nd dominating sets. New values and bounds: (A) (Qn )= dn=2e is shown for 17 values of n (in particular, the set of values for which the conjecture (Q4k+1 )=2 k + 1 is known to hold is extended tok 32); (B) i(Qn )= dn=2e is shown for 11 values of n, including 5 of those from (A); (C) One or both of (Qn )a ndi(Qn) is shown to lie in fdn=2e, dn=2e +1 g for 85 values of n distinct from those in (A) and (B). Combined with previously published work, these results imply that for n 120, each of (Qn )a ndi(Qn) is either known, or known to have one of two values. Also, the general bounds (Qn) 69n=133 + O(1) and i(Qn) 61n=111 + O(1) are established.
TL;DR: The exponential generating function of the proportion of permutations of the symmetric group S_n that admit an $m$-th root is shown.
Abstract: Let $m$ be a positive integer, and $p_n(m)$ the proportion of permutations of the symmetric group $S_n$ that admit an $m$-th root. Calculating the exponential generating function of these permutations, we show the following asymptotic formula $$p_n(m)\, \sim \, {{\pi _m}\over {n^{1-\varphi (m)/m}}},\;\; n\to \infty ,$$ where $\varphi$ is the Euler function and $\pi _m$ an explicit constant.
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TL;DR: A solution for both impartial and partizan versions of the game of End-Nim is given and the signicance of the mystic hieroglyphs are explained.
Abstract: In the game of End-Nim two players take turns in removing one or more boxes from a string of non-empty stacks. At each move boxes may only be taken from the two stacks which form the ends of the string (unless only one stack remains!). We give a solution for both impartial and partizan versions of the game and explain the signicance of the mystic hieroglyphs:
TL;DR: The results include determining $f_1^k(K_n)$, which is equivalent to determining classical Ramsey numbers for multicolorings, and a construction due to Calkin implies that $f-3^k (K-n) \le {{n}\over {k-1}} + k-1$ when $k- 1$ is a prime power.
Abstract: Given a graph $G$ and positive integers $d,k$, let $f_d^k(G)$ be the maximum $t$ such that every $k$-coloring of $E(G)$ yields a monochromatic subgraph with diameter at most $d$ on at least $t$ vertices. Determining $f_1^k(K_n)$ is equivalent to determining classical Ramsey numbers for multicolorings. Our results include $\bullet$ determining $f_d^k(K_{a,b})$ within 1 for all $d,k,a,b$ $\bullet$ for $d \ge 4$, $f_d^3(K_n)=\lceil n/2 \rceil +1$ or $\lceil n/2 \rceil$ depending on whether $n \equiv 2 (mod 4)$ or not $\bullet$ $f_3^k(K_n) > {{n}\over {k-1+1/k}}$ The third result is almost sharp, since a construction due to Calkin implies that $f_3^k(K_n) \le {{n}\over {k-1}} +k-1$ when $k-1$ is a prime power. The asymptotics for $f_d^k(K_n)$ remain open when $d=k=3$ and when $d\ge 3, k \ge 4$ are fixed.
TL;DR: The bipartite Ramsey number b(m,n) is the smallest positive integer such that every (red, green) coloring of the edges of $r$ contains either a red or a green.
Abstract: The bipartite Ramsey number $b(m,n)$ is the smallest positive integer $r$ such that every (red, green) coloring of the edges of $K_{r,r}$ contains either a red $K_{m,m}$ or a green $K_{n,n}$. We obtain asymptotic bounds for $b(m,n)$ for $m \geq 2$ fixed and $n \rightarrow \infty$.
TL;DR: This work builds on work of Boris Pittel concerning the number of $t$-tuples of partitions whose meet (join) is the minimal (maximal) element in the lattice of set partitions.
Abstract: We build on work of Boris Pittel [5] concerning the number of $t$-tuples of partitions whose meet (join) is the minimal (maximal) element in the lattice of set partitions.
TL;DR: The chromatic spectrum of G is the set of all vertices of G for which G has a strict $k$-coloring and it is shown that a set of integers S is the spectrum of some plane graph with face-constraints if and only if $S$ is an interval with a gap.
Abstract: Let $G$ be a plane graph whose vertices are to be colored subject to constraints on some of the faces. There are 3 types of constraints: a $C$ indicates that the face must contain two vertices of a $C$ommon color, a $D$ that it must contain two vertices of a $D$ifferent color and a $B$ that $B$oth conditions must hold simultaneously. A coloring of the vertices of $G$ satisfying the facial constraints is a strict $k$-coloring if it uses exactly $k$ colors. The chromatic spectrum of $G$ is the set of all $k$ for which $G$ has a strict $k$-coloring. We show that a set of integers $S$ is the spectrum of some plane graph with face-constraints if and only if $S$ is an interval $\{s,s+1,\dots,t\}$ with $1\leq s\leq 4$, or $S=\{2,4,5,\dots,t\}$, i.e. there is a gap at 3.
TL;DR: In this paper, the authors present a method for bijective proofs for determinant identities, which is based on translating determinants to Schur functions by the Jacobi-Trudi identity.
Abstract: We present a "method" for bijective proofs for determinant identities, which is based on translating determinants to Schur functions by the Jacobi–Trudi identity. We illustrate this "method" by generalizing a bijective construction (which was first used by Goulden) to a class of Schur function identities, from which we shall obtain bijective proofs for Dodgson's condensation formula, Plucker relations and a recent identity of the second author.
TL;DR: It is proved that if P(x) is a polynomial with nonnegative nondecreasing coefficients and n is a positive integer, then $P(x+n)$ is unimodal.
Abstract: We prove that if $P(x)$ is a polynomial with nonnegative nondecreasing coefficients and $n$ is a positive integer, then $P(x+n)$ is unimodal Applications and open problems are presented
TL;DR: Here I give a proof that provides a uniform combinatorial interpretation of T(m,n) that applies regardless of the sign of $n$.
Abstract: Let $T(m,n)$ denote the number of ways to tile an $m$-by-$n$ rectangle with dominos. For any fixed $m$, the numbers $T(m,n)$ satisfy a linear recurrence relation, and so may be extrapolated to negative values of $n$; these extrapolated values satisfy the relation $$T(m,-2-n)=\epsilon_{m,n}T(m,n),$$ where $\epsilon_{m,n}=-1$ if $m \equiv 2$ (mod 4) and $n$ is odd and where $\epsilon_{m,n}=+1$ otherwise. This is equivalent to a fact demonstrated by Stanley using algebraic methods. Here I give a proof that provides, among other things, a uniform combinatorial interpretation of $T(m,n)$ that applies regardless of the sign of $n$.
TL;DR: It is proven that the feasible set of any planar mixed hypergraph without edges of size two and with an edge of size at least four is gap-free and it is proved that a planar 3-regular bridgeless graph contains a $2$-factor with at least a given number of components.
Abstract: A mixed hypergraph $H$ is a triple $(V,{\cal C},{\cal D})$ where $V$ is its vertex set and ${\cal C}$ and ${\cal D}$ are families of subsets of $V$, ${\cal C}$–edges and ${\cal D}$–edges. A mixed hypergraph is a bihypergraph iff ${\cal C}={\cal D}$. A hypergraph is planar if its bipartite incidence graph is planar. A vertex coloring of $H$ is proper if each ${\cal C}$–edge contains two vertices with the same color and each ${\cal D}$–edge contains two vertices with different colors. The set of all $k$'s for which there exists a proper coloring using exactly $k$ colors is the feasible set of $H$; the feasible set is called gap-free if it is an interval. The minimum (maximum) number of the feasible set is called a lower (upper) chromatic number. We prove that the feasible set of any planar mixed hypergraph without edges of size two and with an edge of size at least four is gap-free. We further prove that a planar mixed hypergraph with at most two ${\cal D}$–edges of size two is two-colorable. We describe a polynomial-time algorithm to decide whether the lower chromatic number of a planar mixed hypergraph equals two. We prove that it is NP-complete to find the upper chromatic number of a mixed hypergraph even for 3-uniform planar bihypergraphs. In order to prove the latter statement, we prove that it is NP-complete to determine whether a planar 3-regular bridgeless graph contains a $2$-factor with at least a given number of components.
TL;DR: A recurrence relation is derived for the rank (over most fields) of the set-inclusion matrices on a finite ground set.
Abstract: A recurrence relation is derived for the rank (over most fields) of the set-inclusion matrices on a finite ground set.
TL;DR: This work shows that apart from a single exceptional class of graphs, if $\sum_{x\in I}\deg(x)\ge n-1$ for every $k$-element independent set, then ${\bf G}$ has a spanning caterpillar with maximum degree $k$.
Abstract: Dirac's classic theorem asserts that if ${\bf G}$ is a graph on $n$ vertices, and $\delta({\bf G})\ge n/2$, then ${\bf G}$ has a hamilton cycle. As is well known, the proof also shows that if $\deg(x)+\deg(y)\ge(n-1)$, for every pair $x$, $y$ of independent vertices in ${\bf G}$, then ${\bf G}$ has a hamilton path. More generally, S. Win has shown that if $k\ge 2$, ${\bf G}$ is connected and $\sum_{x\in I}\deg(x)\ge n-1$ whenever $I$ is a $k$-element independent set, then ${\bf G}$ has a spanning tree ${\bf T}$ with $\Delta({\bf T})\le k$. Here we are interested in the structure of spanning trees under the additional assumption that ${\bf G}$ does not have a spanning tree with maximum degree less than $k$. We show that apart from a single exceptional class of graphs, if $\sum_{x\in I}\deg(x)\ge n-1$ for every $k$-element independent set, then ${\bf G}$ has a spanning caterpillar ${\bf T}$ with maximum degree $k$. Furthermore, given a maximum path $P$ in ${\bf G}$, we may require that $P$ is the spine of ${\bf T}$ and that the set of all vertices whose degree in ${\bf T}$ is $3$ or larger is independent in ${\bf T}$.
TL;DR: The family of all q-element subsets of X can be partitioned into disjoint pairs and the theorem holding for any two such pairs fA1;B1g and fA2;B2g holds is a sharpening of a theorem in [2].
Abstract: Let n and q be given integers and X a nite set with n elements. The following theorem is proved for n>n 0(q). The family of all q-element subsets of X can be partitioned into disjoint pairs (except possibly one if n is odd), so that jA1\A2j+ jB1 \ B2 jq, jA1 \ B2j + jB1 \ A2 jq holds for any two such pairs fA1;B1g and fA2;B2g. This is a sharpening of a theorem in [2]. It is also shown that this is a coding type problem, and several problems of similar nature are posed.