This paper is concerned with the partition of edges of the complete graph Kn and the complete multipartite graph K-m, ... ,-m into subgraphs isomor-phic to cycles. We show that Kn and K-m, ... ,m can be decomposed into certain families of cycles by defining a special decomposition which we call a root path decomposition.

Analyzing pseudo-telepathy graph games, we propose a way to build contextuality scenarios exhibiting the quantum supremacy using graph states. We consider the combinatorial structures generating equivalent scenarios. We investigate which scenarios are more multipartite and show that there exist graphs generating scenarios with a linear multipartiteness width.

We determine necessary and sufficient conditions for a complete multipartite graph to admit a set of 1-factors whose union is the whole graph and, when these conditions are satisfied, we determine the minimum size of such a set. MSC 2000: Primary 05C70; Secondary 05C15.

We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to emergent multipartite structure, separated by a phase transition from a region of disordered graphs. An essential feature is the formalism of graph limits as developed by Lovász et al for dense random graphs.

The f -tolerant arc-forwarding index for a class of complete multipartite graphs are determined by constructing relevant fault tolerant routings. Furthermore, these routings are leveled for Cocktail-party graph and the complete 3-partite graph. Mathematics Subject Classification: 05C38, 05C90, 90B10

How many edges can there be in a maximum matching in a complete multipartite graph? Several cases where the answer is known are discussed, and then a new formula is given which answers this question.