scholarly journals On the Book Thickness of k-Trees

2011 ◽  
Vol Vol. 13 no. 3 (Graph and Algorithms) ◽  
Author(s):  
Vida Dujmović ◽  
David R. Wood
Keyword(s):  
Open Problem ◽  
Discrete Math ◽  
Tree Decomposition ◽  
International Audience

Graphs and Algorithms International audience Every k-tree has book thickness at most k + 1, and this bound is best possible for all k \textgreater= 3. Vandenbussche et al. [SIAM J. Discrete Math., 2009] proved that every k-tree that has a smooth degree-3 tree decomposition with width k has book thickness at most k. We prove this result is best possible for k \textgreater= 4, by constructing a k-tree with book thickness k + 1 that has a smooth degree-4 tree decomposition with width k. This solves an open problem of Vandenbussche et al.

scholarly journals Tiling Periodicity

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Juhani Karhumaki ◽  
Yury Lifshits ◽  
Wojciech Rytter
Keyword(s):  
Open Problem ◽  
Classical Case ◽  
Side Product ◽  
Compact Representation ◽  
Minimal Size ◽  
Partial Word ◽  
International Audience ◽  
New Type

International audience We contribute to combinatorics and algorithmics of words by introducing new types of periodicities in words. A tiling period of a word w is partial word u such that w can be decomposed into several disjoint parallel copies of u, e.g. a lozenge b is a tiling period of a a b b. We investigate properties of tiling periodicities and design an algorithm working in O(n log (n) log log (n)) time which finds a tiling period of minimal size, the number of such minimal periods and their compact representation. The combinatorics of tiling periods differs significantly from that for classical full periods, for example unlike the classical case the same word can have many different primitive tiling periods. We consider also a related new type of periods called in the paper multi-periods. As a side product of the paper we solve an open problem posted by T. Harju (2003).

scholarly journals Separating the k-party communication complexity hierarchy: an application of the Zarankiewicz problem

2011 ◽  
Vol Vol. 13 no. 4 ◽  
Author(s):  
Thomas P. Hayes
Keyword(s):  
Positive Integer ◽  
Open Problem ◽  
Communication Complexity ◽  
60Th Birthday ◽  
Equal Size ◽  
Special Issue ◽  
Algorithms And Complexity ◽  
International Audience ◽  
Zarankiewicz Problem ◽  
Complexity Hierarchy

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For every positive integer k, we construct an explicit family of functions f : \0, 1\(n) -\textgreater \0, 1\ which has (k + 1) - party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Omega (n/k(4)2(k)) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of K. Zarankiewicz, namely, to find the maximum number of edges in a graph on n vertices that does not contain K-s,K-t as a subgraph.

scholarly journals Karp-Miller Trees for a Branching Extension of VASS

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Kumar Neeraj Verma ◽  
Jean Goubault-Larrecq
Keyword(s):  
Open Problem ◽  
Linear Logic ◽  
Tree Automata ◽  
Vector Addition ◽  
International Audience ◽  
Independent Work

International audience We study BVASS (Branching VASS) which extend VASS (Vector Addition Systems with States) by allowing addition transitions that merge two configurations. Runs in BVASS are tree-like structures instead of linear ones as for VASS. We show that the construction of Karp-Miller trees for VASS can be extended to BVASS. This entails that the coverability set for BVASS is computable. This allows us to obtain decidability results for certain classes of equational tree automata with an associative-commutative symbol. Recent independent work by de Groote et al. implies that decidability of reachability in BVASS is equivalent to decidability of provability in MELL (multiplicative exponential linear logic), which is still an open problem. Hence our results are also a step towards answering this question in the affirmative.

scholarly journals A variant of Niessen’s problem on degreesequences of graphs

2014 ◽  
Vol Vol. 16 no. 1 (Graph Theory) ◽  
Author(s):  
Jiyun Guo ◽  
Jianhua Yin
Keyword(s):  
Graph Theory ◽  
Discrete Math ◽  
Simple Graph ◽  
The Other ◽  
Degree Sequences ◽  
International Audience

Graph Theory International audience Let (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) be two sequences of nonnegative integers satisfying the condition that b1>=b2>=...>=bn, ai<= bi for i=1,2,\textellipsis,n and ai+bi>=ai+1+bi+1 for i=1,2,\textellipsis, n-1. In this paper, we give two different conditions, one of which is sufficient and the other one necessary, for the sequences (a1,a2,\textellipsis,an) and (b1,b2,\textellipsis,bn) such that for every (c1,c2,\textellipsis,cn) with ai<=ci<=bi for i=1,2,\textellipsis,n and &#x2211;&limits;i=1n ci=0 (mod 2), there exists a simple graph G with vertices v1,v2,\textellipsis,vn such that dG(vi)=ci for i=1,2,\textellipsis,n. This is a variant of Niessen\textquoterights problem on degree sequences of graphs (Discrete Math., 191 (1998), 247&#x2013;253).

scholarly journals Fast separation in a graph with an excluded minor

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Bruce Reed ◽  
David R. Wood
Keyword(s):  
Open Problem ◽  
Time Algorithm ◽  
Total Weight ◽  
Running Time ◽  
Fast Separation ◽  
Edge Graph ◽  
Excluded Minor ◽  
International Audience ◽  
Vertex Sets

International audience Let $G$ be an $n$-vertex $m$-edge graph with weighted vertices. A pair of vertex sets $A,B \subseteq V(G)$ is a $\frac{2}{3} - \textit{separation}$ of $\textit{order}$ $|A \cap B|$ if $A \cup B = V(G)$, there is no edge between $A \backslash B$ and $B \backslash A$, and both $A \backslash B$ and $B \backslash A$ have weight at most $\frac{2}{3}$ the total weight of $G$. Let $\ell \in \mathbb{Z}^+$ be fixed. Alon, Seymour and Thomas [$\textit{J. Amer. Math. Soc.}$ 1990] presented an algorithm that in $\mathcal{O}(n^{1/2}m)$ time, either outputs a $K_\ell$-minor of $G$, or a separation of $G$ of order $\mathcal{O}(n^{1/2})$. Whether there is a $\mathcal{O}(n+m)$ time algorithm for this theorem was left as open problem. In this paper, we obtain a $\mathcal{O}(n+m)$ time algorithm at the expense of $\mathcal{O}(n^{2/3})$ separator. Moreover, our algorithm exhibits a tradeoff between running time and the order of the separator. In particular, for any given $\epsilon \in [0,\frac{1}{2}]$, our algorithm either outputs a $K_\ell$-minor of $G$, or a separation of $G$ with order $\mathcal{O}(n^{(2-\epsilon )/3})$ in $\mathcal{O}(n^{1+\epsilon} +m)$ time.

scholarly journals q-Enumeration of words by their total variation

2011 ◽  
Vol Vol. 12 no. 3 (Combinatorics) ◽  
Author(s):  
Ligia Loreta Cristea ◽  
Helmut Prodinger
Keyword(s):  
Computer Science ◽  
Recent Work ◽  
Total Variation ◽  
Discrete Math ◽  
International Audience

Combinatorics International audience In recent work, Mansour [Discrete Math. Theoret. Computer Science 11, 2009, 173--186] considers the problem of enumerating all words of length n over {1,2,...,k} (where k is a given integer), that have the total variation equal to a given integer m. In the present paper we study various types of random words over the infinite alphabet ℕ, where the letters have geometric probabilities. We are interested in the probabilities of words of given type to have a given total variation.

scholarly journals On the support of the free Lie algebra: the Schutzenberger problems

2010 ◽  
Vol Vol. 12 no. 3 (Combinatorics) ◽  
Author(s):  
Ioannis C. Michos
Keyword(s):  
Lie Algebra ◽  
Recursion Formula ◽  
Discrete Math ◽  
Binomial Coefficient ◽  
Finite Alphabet ◽  
Free Lie Algebra ◽  
Lie Ring ◽  
Linear Polynomial ◽  
International Audience ◽  
Natural Way

Combinatorics International audience M.-P. Schutzenberger asked to determine the support of the free Lie algebra L(Zm) (A) on a finite alphabet A over the ring Z(m) of integers mod m and all pairs of twin and anti-twin words, i.e., words that appear with equal (resp. opposite) coefficients in each Lie polynomial. We characterize the complement of the support of L(Zm) (A) in A* as the set of all words w such that m divides all the coefficients appearing in the monomials of l* (w), where l* is the adjoint endomorphism of the left normed Lie bracketing l of the free Lie ring. Calculating l* (w) via the shuffle product, we recover the well known result of Duchamp and Thibon (Discrete Math. 76 (1989) 123-132) for the support of the free Lie ring in a much more natural way. We conjecture that two words u and v of common length n, which lie in the support of the free Lie ring, are twin (resp. anti-twin) if and only if either u = v or n is odd and u = (v) over tilde (resp. if n is even and u = (v) over tilde), where (v) over tilde denotes the reversal of v and we prove that it suffices to show this for a two-lettered alphabet. These problems can be rephrased, for words of length n, in terms of the action of the Dynkin operator l(n) on lambda-tabloids, where lambda is a partition of n. Representing a word w in two letters by the subset I of [n] = \1, 2, ... , n\ that consists of all positions that one of the letters occurs in w, the computation of l* (w) leads us to the notion of the Pascal descent polynomial p(n)(I), a particular commutative multi-linear polynomial which is equal to the signed binomial coefficient when vertical bar I vertical bar = 1. We provide a recursion formula for p(n) (I) and show that if m inverted iota Sigma(i is an element of I)(1)(i-1) (n - 1 i - 1), then w lies in the support of L(Zm) (A).

scholarly journals Combinatorial Reciprocity for Monotone Triangles

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Ilse Fischer ◽  
Lukas Riegler
Keyword(s):  
Open Problem ◽  
Alternating Sign Matrices ◽  
Combinatorial Objects ◽  
Bijective Proof ◽  
International Audience ◽  
One To One

International audience The number of Monotone Triangles with bottom row $k_1 < k_2 < ⋯< k_n$ is given by a polynomial $\alpha (n; k_1,\ldots,k_n)$ in $n$ variables. The evaluation of this polynomial at weakly decreasing sequences $k_1 ≥k_2 ≥⋯≥k_n $turns out to be interpretable as signed enumeration of new combinatorial objects called Decreasing Monotone Triangles. There exist surprising connections between the two classes of objects – in particular it is shown that $\alpha (n;1,2,\ldots,n) = \alpha (2n; n,n,n-1,n-1,\ldots,1,1)$. In perfect analogy to the correspondence between Monotone Triangles and Alternating Sign Matrices, the set of Decreasing Monotone Triangles with bottom row $(n,n,n-1,n-1,\ldots,1,1)$ is in one-to-one correspondence with a certain set of ASM-like matrices, which also play an important role in proving the claimed identity algebraically. Finding a bijective proof remains an open problem. Le nombre de Triangles Monotones ayant pour dernière ligne $k_1 < k_2 < ⋯< k_n$ est donné par un polynôme $\alpha (n; k_1,\ldots,k_n)$ en $n$ variables. Il se trouve que les valeurs de ce polynôme en les suites décroissantes $k_1 ≥k_2 ≥⋯≥k_n$ peuvent s'interpréter comme l'énumération signée de nouveaux objets appelés Triangles Monotones Décroissants. Il existe des liens surprenants entre ces deux classes d'objets – en particulier on prouvera l'identité $\alpha (n;1,2,\ldots,n) = \alpha (2n; n,n,n-1,n-1,\ldots,1,1)$. En parfaite analogie avec la correspondance entre Triangles Monotones et Matrices à Signe Alternant, l'ensemble des Triangles Monotones Décroissants ayant pour dernière ligne $(n,n,n-1,n-1,\ldots,1,1)$ est en correspondance biunivoque avec un certain ensemble de matrices similaires aux MSAs, ce qui joue un rôle important dans la preuve algébrique de l'identité précédente. C'est un problème ouvert que d'en donner une preuve bijective.

scholarly journals On an open problem of Green and Losonczy: exact enumeration of freely braided permutations

2004 ◽  
Vol Vol. 6 no. 2 ◽  
Author(s):  
Toufik Mansour
Keyword(s):  
Representation Theory ◽  
Generating Function ◽  
Coxeter Group ◽  
Upper Bound ◽  
Open Problem ◽  
Special Class ◽  
Exact Enumeration ◽  
Restricted Permutations ◽  
International Audience ◽  
Freely Braided

International audience Recently, Green and Losonczy~GL1,GL2 introduced \emphfreely braided permutation as a special class of restricted permutations has arisen in representation theory. The freely braided permutations were introduced and studied as the upper bound for the number of commutation classes of reduced expressions for an element of a simply laced Coxeter group is achieved if and only if when the element is freely braided. In this paper, we prove that the generating function for the number of freely braided permutations in S_n is given by \par (1-3x-2x^2+(1+x)√1-4x) / (1-4x-x^2+(1-x^2)√1-4x).\par

scholarly journals Separation dimension and degree

2019 ◽  
pp. 1-10
Author(s):  
ALEX SCOTT ◽  
DAVID R. WOOD
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Chromatic Number ◽  
Open Problem ◽  
Maximum Degree ◽  
Discrete Math ◽  
Constant Factor ◽  
Axis Parallel ◽  
Parallel Hyperplane ◽  
Separation Dimension

Abstract The separation dimension of a graph G is the minimum positive integer d for which there is an embedding of G into ℝ d , such that every pair of disjoint edges are separated by some axis-parallel hyperplane. We prove a conjecture of Alon et al. [SIAM J. Discrete Math. 2015] by showing that every graph with maximum degree Δ has separation dimension less than 20Δ, which is best possible up to a constant factor. We also prove that graphs with separation dimension 3 have bounded average degree and bounded chromatic number, partially resolving an open problem by Alon et al. [J. Graph Theory 2018].

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