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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]

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]

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]

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]

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]

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]