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Unformatted text preview: 10.10 Additional Topics on Cycles 583 Proposition 10.9.7 If n = gs , g ≥ 2 , then there exists an s-regular round digraph of order n which is s-strong and has girth g . ut 10.10 Additional Topics on Cycles 10.10.1 Chords of Cycles The existence of chords of cycles is not only an interesting problem by itself, it has also several applications. One of these applications is the existence of kernels in digraphs (see Subsection 12.3.1), another one will be described in this subsection. Let D be a directed multigraph with δ ( D ) ≥ k . It is not difficult to see that D has a cycle with at least k- 1 chords. Indeed, let P = p 1 p 2 ...p k be a longest path in D . Clearly, there are k arcs from p k to vertices of P . These arcs and part of P form the desired cycle with k- 1 chords. While for k = 1 this result cannot be improved (consider ~ C n or ‘tree-like’ strong digraphs obtained from several cycles in such a way that every pair of cycles has at most one common vertex). Marcus  showed that for k ≥ 2 the above simple result can be improved to the following: Theorem 10.10.1 (Marcus’ theorem)  Let D be a strong directed multigraph with at least two vertices and δ ( D ) ≥ k ≥ 2 . Then D contains a cycle with at least k chords. ut This result improves and extends the main assertion by Thomassen  that every 2-arc-strong directed multigraph has a cycle with at least two chords. The proof of Theorem 10.10.1 in  is quite involved and lengthy, and thus is not given here. Instead, we will consider an interesting applica- tion of Theorem 10.10.1 to the problem of minimum size strong spanning subgraphs of strong directed multigraphs (often called the minimum equiva- lent subdigraph problem, see the end of Section 4.3 and Section 6.11). Lemma 10.10.2  Let k be a positive integer, let a and b be non-negative real numbers, and suppose that every k-arc-strong directed multigraph with at least two vertices has a strong subgraph H with at least two vertices and a strong factor 9 H of H such that e ≤ ae + b ( h- 1) , where h is the order of H and e ( e ) is the size of H ( H ). Then every k- arc-strong directed multigraph of order n and size m has a strong factor with at most am + b ( n- 1) arcs. 9 Recall that a factor is a spanning subdigraph. 584 10. Cycle Structure of Digraphs Proof: This holds trivially for directed multigraphs with one vertex since a ≥ . Thus, consider a directed multigraph D of order n ≥ 2 and assume that the result is true for all directed multigraphs with less than n vertices. By the assumption, D has a subgraph H as in the lemma. Clearly the contracted directed multigraph D/H is k-arc-strong and has n- h + 1 < n vertices; so D/H has a factor with a ( m- e )+ b ( n- h ) arcs. The corresponding arcs of D , along with the e arcs of H , form a factor of D of size at most am + b ( n- 1) ....
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This document was uploaded on 08/10/2011.
- Spring '11