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Given a set system (also well-known as a hypergraph) H = {U, X}, where U is a set of elements and X is a set of subsets of U, an exact hitting set S is a subset of U such that each subset in X contains exactly one element in S. We refer to a set system as exactly hittable if it has an exact hitting set. In this paper, we study interval graphs which have intersection models that are exactly hittable. We refer to these interval graphs as Exactly Hittable Interval Graphs (EHIG). We present a forbidden structure characterization for EHIG. We also show that the class of proper interval graphs is a strict subclass of EHIG. Finally, we give an algorithm that runs in polynomial time to recognize graphs belonging to the class of EHIG. [ABSTRACT FROM AUTHOR]

We define a q-linear path in a hypergraph H as a sequence (e1,. . ., eL) of edges of H such that |ei nei+1| [[1, q]] and ei n ej = Ø if |i-j| > 1. In this paper, we study the connected components associated to these paths when q = k - 2 where k is the rank of H. If k = 3 then q = 1 which coincides with the well-known notion of linear path or loose path. We describe the structure of the connected components, using an algorithmic proof which shows that the connected components can be computed in polynomial time. We then mention two consequences of our algorithmic result. The first one is that deciding the winner of the Maker-Breaker game on a hypergraph of rank 3 can be done in polynomial time. The second one is that tractable cases for the NP-complete problem of "Paths Avoiding Forbidden Pairs" in a graph can be deduced from the recognition of a special type of line graph of a hypergraph. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of gossiping with interference constraint in radio ring networks. Gossiping (or total exchange information) is a protocol where each node in the network has a message and is expected to distribute its own message to every other node in the network. The gossiping problem consists in finding the minimum running time (makespan) of a gossiping protocol and algorithms that attain this makespan. We focus on the case where the transmission network is a ring network. We consider synchronous protocols where it takes one unit of time (step) to transmit a unit-length message. During one step, a node receives at most one message only through one of its two neighbors. We also suppose that, during one step, a node cannot be both a sender and a receiver (half duplex model). Moreover communication is subject to interference constraints. We use a primary node interference model where, if a node receives a message from one of its neighbors, its other neighbor cannot send at the same time. With these assumptions we completely solve the problem for ring networks. We first show lower bounds and then give gossiping algorithms which meet these lower bounds and so are optimal. The number of rounds depends on the congruences of n modulo 12. [ABSTRACT FROM AUTHOR]

The (bitwise) complement x of a binary word x is obtained by changing each 0 in x to 1 and vice versa. An antisquare is a nonempty word of the form x x. In this paper, we study infinite binary words that do not contain arbitrarily large antisquares. For example, we show that the repetition threshold for the language of infinite binary words containing exactly two distinct antisquares is (5 +√5)/2. We also study repetition thresholds for related classes, where "two" in the previous sentence is replaced by a larger number. We say a binary word is good if the only antisquares it contains are 01 and 10. We characterize the minimal antisquares, that is, those words that are antisquares but all proper factors are good. We determine the growth rate of the number of good words of length n and determine the repetition threshold between polynomial and exponential growth for the number of good words. [ABSTRACT FROM AUTHOR]

We generalize the familiar notion of periodicity in sequences to a new kind of pseudoperiodicity, and we prove some basic results about it. We revisit the results of a 2012 paper of Shevelev and reprove his results in a simpler and more unified manner, and provide a complete answer to one of his previously unresolved questions. We consider finding words with specific pseudoperiod and having the smallest possible critical exponent. Finally, we consider the problem of determining whether a finite word is pseudoperiodic of a given size, and show that it is NP-complete. [ABSTRACT FROM AUTHOR]

Mixed graphs have both directed and undirected edges. We consider mixed graphs that are regular in both directed and undirected degree and with given girth. Araujo-Pardo, Hern'andez-Cruz, andMontellano-Ballesteros recently established a lower bound for regular mixed graphs with directed degree one that generalizes the Moore bound. In this paper we show that, in at least one non-trivial case, their bound can be achieved. For a few other specific values of direct degree, undirected degree and girth we are able to identify graphs (mixed cages) of minimum possible order. We also obtain some fairly tight bounds for a few other cases. [ABSTRACT FROM AUTHOR]

The analysis of strings of n random variables with geometric distribution has recently attracted renewed interest: Archibald et al. consider the number of distinct adjacent pairs in geometrically distributed words. They obtain the asymptotic (n → ∞) mean of this number in the cases of different and identical pairs. In this paper we are interested in all asymptotic moments in the identical case, in the asymptotic variance in the different case and in the asymptotic distribution in both cases. We use two approaches: the first one, the probabilistic approach, leads to variances in both cases and to some conjectures on all moments in the identical case and on the distribution in both cases. The second approach, the combinatorial one, relies on multivariate pattern matching techniques, yielding exact formulas for first and second moments. We use such tools as Mellin transforms, Analytic Combinatorics, Markov Chains. [ABSTRACT FROM AUTHOR]

A graph G is weighted–k–list–antimagic if for any vertex weighting ω: V (G) → R and any list assignment L: E(G) → 2R with |L(e)| ≥ |E(G)| + k there exists an edge labeling f such that f(e) ∈ L(e) for all e ∈ E(G), labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on n vertices having no K1 or K2 component is weighted–[ 4n/3 ] –list–antimagic. [ABSTRACT FROM AUTHOR]

The Tanny sequence T(i) is a sequence defined recursively as T(i) = T(i - 1 - T(i - 1)) + T(i - 2 - T(i - 2)), T(0) = T(1) = T(2) = 1. In the first part of this paper we give combinatorial proofs of all the results regarding T(i), that Tanny proved in his paper "A well-behaved cousin of the Hofstadter sequence", Discrete Mathematics, 105(1992), pp. 227-239, using algebraic means. In most cases our proofs turn out to be simpler and shorter. Moreover, they give a "visual" appeal to the theory developed by Tanny. We also generalize most of Tanny's results. In the second part of the paper we present many new results regarding T(i) and prove them combinatorially. Given two integers n and k, it is interesting to know if T(n) = k or not. In this paper we characterize such numbers. [ABSTRACT FROM AUTHOR]

In 1961, Erdős asked whether or not there exist words of arbitrary length over a fixed finite alphabet that avoid patterns of the form XX' where X' is a permutation of X (called abelian squares). This problem has since been solved in the affirmative in a series of papers from 1968 to 1992. Much less is known in the case of abelian k-th powers, i.e., words of the form X1X2…Xk where Xi is a permutation of X1 for 2 ≤ i ≤ k. In this paper, we consider crucial words for abelian k-th powers, i.e., finite words that avoid abelian k-th powers, but which cannot be extended to the right by any letter of their own alphabets without creating an abelian k-th power. More specifically, we consider the problem of determining the minimal length of a crucial word avoiding abelian k-th powers. This problem has already been solved for abelian squares by Evdokimov and Kitaev (2004), who showed that a minimal crucial word over an n-letter alphabet An {1,2,…,n} avoiding abelian squares has length 4n-7 for n ≥ 3. Extending this result, we prove that a minimal crucial word over An avoiding abelian cubes has length 9n - 13 for n ≥ 5, and it has length 2, 5, 11, and 20 for n = 1, 2, 3, and 4, respectively. Moreover, for n ≥ 4 and k ge; 2, we give a construction of length k²(n -- 1) - k - 1 of a crucial word over An avoiding abelian k-th powers. This construction gives the minimal length for k = 2 and k = 3. For k ≥ 4 and n ≥ 5, we provide a lower bound for the length of crucial words over An avoiding abelian k-th powers. [ABSTRACT FROM AUTHOR]

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 ◊ b is a tiling period of aabb. 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). [ABSTRACT FROM AUTHOR]

In this paper, we study the residual solvability of the generalized free product of solvable groups. [ABSTRACT FROM AUTHOR]

It is known that a Sturmian sequence S can be defined as a coding of the orbit of ρ (called the intercept of S) under a rotation of irrational angle a (called the slope). On the other hand, a fixed point of an invertible substitution is Sturmian. Naturally, there are two interrelated questions: (1) Given an invertible substitution, we know that its fixed point is Sturmian. What is the slope and intercept? (2) Which kind of Sturmian sequences can be fixed by certain non-trivial invertible substitutions? In this paper we give a unified treatment to the two questions. We remark that though the results are known, our proof is very elementary and concise. [ABSTRACT FROM AUTHOR]

The split-coloring problem is a generalized vertex coloring problem where we partition the vertices into a minimum number of split graphs. In this paper, we study some notions which are extensively studied for the usual vertex coloring and the cocoloring problem from the point of view of split-coloring, such as criticality and the uniqueness of the minimum split-coloring. We discuss some properties of split-critical and uniquely split-colorable graphs. We describe constructions of such graphs with some additional properties. We also study the effect of the addition and the removal of some edge sets on the value of the split-chromatic number. All these results are compared with their cochromatic counterparts. We conclude with several research directions on the topic. [ABSTRACT FROM AUTHOR]

A graph is a probe interval graph if its vertices can be partitioned into probes and nonprobes with an interval associated to each vertex so that vertices are adjacent if and only if their corresponding intervals intersect and at least one of them is a probe. A graph G = (V,E) is a tolerance graph if each vertex v ∈ V can be associated to an interval Iv of the real line and a positive real number tv such that uv ∈ E if and only if |Iu ∩ Iv| ≥ min {tu; tv}. In this paper we present O(|V| + |E|) recognition algorithms for both bipartite probe interval graphs and bipartite tolerance graphs. We also give a new structural characterization for each class which follows from the algorithms. [ABSTRACT FROM AUTHOR]

For a graph G and integers a and b, an (a, b)-code of G is a set C of vertices such that any vertex from C has exactly a neighbors in C and any vertex not in C has exactly b neighbors in C. In this paper we classify integers a and b for which there exist (a, b)-codes in Sierpiński graphs. [ABSTRACT FROM AUTHOR]

We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co-multiplication laws, thereby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the concepts of composition and decomposition with the emphasis on combinatorial origin of the ensuing algebraic constructions. [ABSTRACT FROM AUTHOR]

Several matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question "Which graphs are determined by their spectrum?" remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by considering some unicyclic graphs. An odd (resp. even) sun is a graph obtained by appending a pendant vertex to each vertex of an odd (resp. even) cycle. A broken sun is a graph obtained by deleting pendant vertices of a sun. In this paper we prove that a sun is determined by its Laplacian spectrum, an odd sun is determined by its adjacency spectrum (counter-examples are given for even suns) and we give some spectral characterizations of broken suns. [ABSTRACT FROM AUTHOR]

A Parry number is a real number β > 1 such that the Rényi β-expansion of 1 is finite or infinite eventually periodic. If this expansion is finite, β is said to be a simple Parry number. Remind that any Pisot number is a Parry number. In a previous work we have determined the complexity of the fixed point u β of the canonical substitution associated with β-expansions, when β is a simple Parry number. In this paper we consider the case where β is a non-simple Parry number. We determine the structure of infinite left special branches, which are an important tool for the computation of the complexity of uβ. These results allow in particular to obtain the following characterization: the infinite word uβ is Sturmian if and only if β is a quadratic Pisot unit. [ABSTRACT FROM AUTHOR]

At the 4th Conference on Combinatorics on Words, Christophe Reutenauer posed the question of whether the dual reflected order yields a Gray code on the Lyndon family. In this paper we give a positive answer. More precisely, we present an O(1)-average-time algorithm for generating length n binary pre-necklaces, necklaces and Lyndon words in Gray code order. [ABSTRACT FROM AUTHOR]

In a previous paper, we characterized free monoid morphisms preserving finite Lyndon words. In particular, we proved that such a morphism preserves the order on finite words. Here we study morphisms preserving infinite Lyndon words and morphisms preserving the order on infinite words. We characterize them and show relations with morphisms preserving Lyndon words or the order on finite words. We also briefly study morphisms preserving border-free words and those preserving the radix order. [ABSTRACT FROM AUTHOR]

The strong chromatic index of a graph is the minimum number of colours needed to colour the edges in such a way that each colour class is an induced matching. In this paper, we present bounds for the strong chromatic index of three different products of graphs in terms of the strong chromatic index of each factor. For the Cartesian product of paths, cycles or complete graphs, we derive sharper results. In particular, strong chromatic indices of d-dimensional grids and of some toroidal grids are given along with approximate results on the strong chromatic index of generalized hypercubes. [ABSTRACT FROM AUTHOR]

This paper deals with statistics concerning distances between randomly chosen nodes in varieties of increasing trees. Increasing trees are labelled rooted trees where labels along any branch from the root go in increasing order. Many important tree families that have applications in computer science or are used as probabilistic models in various applications, like recursive trees, heap-ordered trees or binary increasing trees (isomorphic to binary search trees) are members of this variety of trees. We consider the parameters depth of a randomly chosen node, distance between two randomly chosen nodes, and the generalisations where p nodes are randomly chosen: the size of the ancestor-tree of these selected nodes and the size of the smallest subtree generated by these nodes, also called Steiner distance. Under the restriction that the node-degrees are bounded, we can prove that all these parameters converge in law to the Normal distribution. This extends results obtained earlier for binary search trees and heap-ordered trees to a much larger class of structures. [ABSTRACT FROM AUTHOR]

Recently, Green and Losonczy [5, 6] introduced freely braided permutations as a special class of restricted permutations that has arisen in the study of Schubert varieties. They suggest as an open problem to enumerate the number of freely braided permutations in Sn. In this paper, we prove that the generating function for the number of freely braided permutations in Sn is given by 1-3x-2x²+(1+x)√p1-4x/1-4x-x²+(1-x²)√1-4x. [ABSTRACT FROM AUTHOR]

The paper addresses the cheating prevention in secret sharing. We consider secret sharing with binary shares. The secret also is binary. This model allows us to use results and constructions from the well developed theory of cryptographically strong boolean functions. In particular, we prove that for given secret sharing, the average cheating probability over all cheating vectors and all original vectors, i.e., 1/n x 2nΣc=1nΣα∈Vnρc,α, denoted by ρ, satisfies ρ ≥ ½, and the equality holds if and only if ρc,&alpha satisfies ρc,α = ½ for every cheating vector δc and every original vector α. In this case the secret sharing is said to be cheating immune. We further establish a relationship between cheating-immune secret sharing and cryptographic criteria of boolean functions. This enables us to construct cheating-immune secret sharing. [ABSTRACT FROM AUTHOR]

Let G be a graph on vertices v1,, v2,..., vn. The b-chromatic number of G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i &;le; k, has at least one representant xi adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we give the exact value for the b-chromatic number of power graphs of a path and we determine bounds for the b-chromatic number of power graphs of a cycle. [ABSTRACT FROM AUTHOR]

Constellations are colored planar maps that generalize different families of maps (planar maps, bipartite planar maps, bi-Eulerian planar maps, planar cacti, . . . ) and are strongly related to factorizations of permutations. They were recently studied by Bousquet-Mélou and Schaeffer [9] who describe a correspondence between these maps and a family of trees, called Eulerian trees. In this paper, we derive from their result a relationship between planar constellations and another family of trees, called stellar trees. This correspondence generalizes a well known result for planar cacti, and shows that planar constellations are colored Lagrangian objects (that is objects that can be enumerated by the Good-Lagrange formula). We then deduce from this result a new formula for the number of planar constellations having a given face color distribution, different from the formula one can derive from the results of Bousquet-Mélou and Schaeffer, along with systems of functional equations for the generating functions of bipartite and bi-Eulerian planar maps enumerated according to the number of vertices of each color and the number of faces. [ABSTRACT FROM AUTHOR]

Let G be a graph on vertices v[sub1], v[sub2],..., v[subn]. The b-chromatic number of G is defined as the maximum number k of colors that can be used to color the vertices of G, such that we obtain a proper coloring and each color i, with 1 ≤ i ≤ k, has at least one representant x[subi] adjacent to a vertex of every color j, 1 ≤ j ≠ i ≤ k. In this paper, we give the exact value for the b-chromatic number of power graphs of a path and we determine bounds for the b-chromatic number of power graphs of a cycle. [ABSTRACT FROM AUTHOR]