Digraphs+Theory,+Algorithms+and+Applications_Part30 - 10.4...

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Unformatted text preview: 10.4 Disjoint Cycles Versus Feedback Sets 563 10.4.2 Solution of Younger’s Conjecture The vertex and arc versions of Younger’s conjecture were proved for various families of digraphs including the families mentioned above. McCuaig [559] proved the existence of t (2) by characterizing intercyclic digraphs , i.e., digraphs D for which ν ( D ) ≤ 1. Moreover, he established that t (2) = 3. Reed and Shepherd [627] proved the vertex version of Younger’s conjecture for planar digraphs using a result of Seymour [665]. The result of Reed and Shepherd combined with a result of Goemans and Williamson [323] implies that t pd ( c ) = O ( c ), where t pd ( c ) is the function t ( c ) restricted to planar di- graphs. Finally, Younger’s conjecture was completely settled by Reed, Robert- son, Seymour and Thomas [626]. In this subsection, we give a scheme of their proof. In particular, we provide a complete proof of perhaps the most inter- esting lemma in [626]. One of the important tools in the proof in [626] is the following well-known Ramsey theorem [621]. Theorem 10.4.5 (Ramsey) For all integers q,l,r ≥ 1 there exists a (min- imum) integer R l ( r,q ) ≥ so that the following holds. Let Z be a set of car- dinality at least R l ( r,q ) and let every l-subset of Z be assigned a colour from { 1 ,...,q } . Then there exist an r-subset S of Z and a colour k ∈ { 1 ,...,q } so that every l-subset of S is of colour k . ut Some readers may be more familiar with the graph-theoretic special case of this theorem. For every pair of natural numbers q,r there exists an integer R 2 ( r,q ) ≥ 0 so that every q-edge-coloured complete graph of order at least R 2 ( r,q ) has a monochromatic complete subgraph of order r . We start describing the scheme of the proof of Younger’s conjecture by the following lemma whose proof is left as Exercise 10.24. Lemma 10.4.6 [626] Let c ≥ 1 be an integer such that t ( c- 1) exists. Let D be a digraph with ν ( D ) < c and let T be a feedback vertex set of D of cardinality τ ( D ) . Suppose U,W are disjoint subsets of T both of cardinality r , where r ≥ 2 t ( c- 1) . Then there is an r-path subdigraph of D from U to W , which contains no vertex in T- ( U ∪ W ) . ut Let L = P 1 ∪ ... ∪ P k be a k-path subdigraph in a digraph D and let u i ( w i ) be the initial (terminal) vertex in P i , i = 1 ,..., 2. We say that L links ( u 1 ,...,u k ) to ( w 1 ,...,w k ) and L is from { u 1 ,...,u k } to { w 1 ,...,w k } . The following lemma was proved by the authors of [626] in joint work with Alon. Its proof uses Ramsey’s theorem as well as Theorem 5.2.3 of Erd˝os and Szekeres. 564 10. Cycle Structure of Digraphs Lemma 10.4.7 Let c ≥ 2 be an integer such that t ( c- 1) exists, and let k ≥ 1 be an integer. Then there exists an integer t ≥ (depending on k ) so that the following holds. If D is a digraph with ν ( D ) < c and τ ( D ) ≥ t , then there are distinct vertices u 1 ,...,u k ,w 1 ,...,w k of D and a pair of...
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Digraphs+Theory,+Algorithms+and+Applications_Part30 - 10.4...

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