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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]

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]

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]