Cycle decompositions IV: complete directed graphs and fixed length directed cycles

Cycle decompositions IV: complete directed graphs and fixed length directed cycles

We establish necessary and sufficient conditions for decomposing the complete symmetric digraph of order n into directed cycles of length m; where 2pmpn: r 2003 Elsevier Science (USA). All rights reserved.

Cycle Decompositions Iii: Complete Graphs and Fixed Length Cycles

Directed 3-cycle decompositions of complete directed graphs with quadratic leaves

Let G be a vertex-disjoint union of directed cycles in the complete directed graph Dt , let |E(G)| be the number of directed edges of G and suppose G 6= ⇀ C 2 ∪ ⇀ C 3 or ⇀ C 5 if t = 5, and G 6= ⇀ C 3 ∪ ⇀ C 3 if t = 6. It is proved in this paper that for each positive integer t , there exist ⇀ C 3-decompositions for Dt − G if and only if t(t − 1)− |E(G)| ≡ 0 (mod 3). © 2008 Elsevier B.V. All ri...

Cycle decompositions V: Complete graphs into cycles of arbitrary lengths

We show that the complete graph on n vertices can be decomposed into t cycles of specified lengths m1, . . . ,mt if and only if n is odd, 3 ≤ mi ≤ n for i = 1, . . . , t, and m1+· · ·+mt = ( n 2 ) . We also show that the complete graph on n vertices can be decomposed into a perfect matching and t cycles of specified lengths m1, . . . ,mt if and only if n is even, 3 ≤ mi ≤ n for i = 1, . . . , t...

Directed graphs, decompositions, and spatial linkages

The decomposition of a linkage into minimal components is a central tool of analysis and synthesis of linkages. In this paper we prove that every pinned d-isostatic (minimally rigid) graph (grounded linkage) has a unique decomposition into minimal strongly connected components (in the sense of directed graphs), or equivalently intominimal pinned isostatic graphs, which we call d-Assur graphs. W...