ISOMORPHIC PATH DECOMPOSITIONS OF $λK_{n,n,n} (λK^{∗}_{n,n,n})$ FOR ODD $n$

Path Decompositions for Markov Chains

We present two path decompositions of Markov chains (with general state space) by means of harmonic functions, which are dual to each other. They can be seen as a generalization of Williams' decomposition of a Brownian motion with drift. The results may be illustrated by a multitude of examples, but we confine ourselves to different types of random walks and the Pólya urn. 1. Introduction and m...

Minimal Decompositions of Hypergraphs into Mutually Isomorphic Subhypergraphs

Path And Cycle Decompositions

First I want to say a few words about my graph terminology. If I want to allow loops, I use the adjective reflexive. If I want to allow multiple edges, I use multigraph. Thus, a graph has no loops and no multiple edges. I use valency rather than degree. If we say a graph is 4-valent (or tetravalent), it means it is regular of valency 4, for example. A decomposition of a graph X is a partition o...

Minimal decompositions of graphs into mutually isomorphic subgraphs

Suppose 9.= !{pGl , . . ., Gk} is a collection of graphs, all having n vertices and e edges. By a U-decomposition of T„ we mean a set of partitions of the edge sets E(G,) of the G,, say E(G,)= = E E,f , such that for each j, all the E,j , 15 is k, are isomorphic as graphs . Define the function U(T„) to be the least possible value of r a U-decomposition of W„ can have . Finally, let Uk (n) denot...

Decompositions of nearly complete digraphs into t isomorphic parts

An arc decomposition of the complete digraph DKn into t isomorphic subdigraphs is generalized to the case where the numerical divisibility condition is not satisfied. Two sets of nearly tth parts are constructively proved to be nonempty. These are the floor tth class (DKn − R)/t and the ceiling tth class (DKn + S)/t, where R and S comprise (possibly copies of) arcs whose number is the smallest ...