New PDF release: A Note on Complete Subdivisions in Digraphs of Large

By Kuhn D., Osthus D.

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3, we have E(Xi ) = Pr(e ∈ EGi (A, Ac )) = e∈Ei mi . 2 Two edges e, f (e = f ) in Ei are said to be linked if there exists p1 , p2 ∈ P such that e ∪ f ⊆ p1 ∪ p2 . For e, f ∈ Ei , we have 1   2 if e, f are linked and not incident; c Pr(e, f ∈ EGi (A, A )) = 0 if e, f are linked and incident;  1 otherwise. 4 For any edge e ∈ Ei , there is at most one edge f ∈ Ei that is linked and not incident to e. Hence, there are at most mi (ordered) pairs of edges of Ei that are linked and not incident. 3, we have E(Xi2 ) = E(Xi ) + Pr(e, f ∈ EGi (A, Ac )) e,f ∈Ei e=f 1 1 1 mi + [mi (mi − 1) − mi ] + mi 2 4 2 1 2 1 = mi + mi , 4 2 ≤ and σi 2 = E(Xi2 ) − E(Xi )2 ≤ 1 m.

We perform induction on separately for (i), (ii). For G = H, (i) is trivially true. Suppose that (i) is true for some specific G and A let C = C(G, D, T ) as there. Consider G+ := G + x where A is a clique of order at most 2 in G. Then A can not intersect more than one component of G − T . Thus, T separates G+ , too. If A does not intersect V (C) then C+ := C is a component of G+ − T , too, and if, otherwise, A does intersect V (C) then C+ := G+ (V (C) ∪ {x}) A is a component of G+ − T and G(V (C+ ) ∪ T ) = H(V (C) ∪ T ) + x holds.

41 42 JOURNAL OF GRAPH THEORY In order to transform a 3-connected graph G into a smaller one, Tutte suggested to delete an edge xy from G and then to suppress the vertices of degree 2 among x, y. The resulting graph, denoted by G − −xy, can contain vertices of degree less than 3 (and this is not the only reason for which it might fail to be 3-connected). However, Tutte proved that either G − −xy or the graph G/xy obtained from G by contracting xy is 3-connected (implicitely in [6]). One might consider a similar sequence of operations which starts with the deletion of a vertex x from some 3-connected graph G and continues with suppressions of its neighbors of degree less than 3.

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A Note on Complete Subdivisions in Digraphs of Large Outdegree by Kuhn D., Osthus D.


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