Decomposition of the complete directed graph into k-circuits

Packing Circuits into K

We shall pack circuits of arbitrary lengths into the complete graph KN . More precisely, if N is odd and ∑t i=1 mi = (N 2 ) , mi ≥ 3, then the edges of KN can be written as an edge disjoint union of circuits of lengths m1, . . . , mt. Since the degrees of the vertices in any such packing must be even, this result cannot hold for even N . For N even, we prove that if ∑t i=1 mi ≤ (N 2 )− N 2 then...

k-optimal Partitions of a Directed Graph

A partition M is k-optimal if M minimizes BdM). For instance, if G has a Hamiltonian path J.Lo, then M = W.o} is a k -optimal partition. So the k -optimal partitions extend in some sense the concept of Hamiltonian paths. The theorem of Greene and Kleitman [10], which extends the Dilworth theorem [5], shows an important property of k-optimal partitions for the graph of a partially ordered set (i...

Finding All the Elementary Circuits of a Directed Graph

An algorithm is presented which finds all the elementary circuits-of a directed graph in time bounded by O((n + e)(c + 1)) and space bounded by O(n + e), where there are n vertices, e edges and c elementary circuits in the graph. The algorithm resembles algorithms by Tiernan and Tarjan, but is faster because it considers each edge at most twice between any one circuit and the next in the output...

Directed prime graph of non-commutative ring

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...

Graph decomposition is NPC - A complete proof