By Kuhn D., Osthus D.
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Extra info for A Note on Complete Subdivisions in Digraphs of Large Outdegree
They avoid x, and their edges together with the edges of C form a subdivision of K4 avoiding x. Now Lemma 2 implies our statement. 1002/jgt 48 JOURNAL OF GRAPH THEORY According to Lemma 6, we should concentrate on graphs without separating cycles. These have been studied thoroughly in . (a), Fig. 1]. One can check that each graph G of them possesses a vertex x such that G − −x is K4 or some supergraph of K3,3 . For any 4-connected graph G, G − −x = s(G − x) = G − x holds since δ(s(G − x)) ≥ 3.
1002/jgt 32 JOURNAL OF GRAPH THEORY ACKNOWLEDGMENTS The author thanks Graham Brightwell and Jan van den Heuvel for helpful comments and discussions. The research work was supported by an EPSRC Doctoral Training Grant. REFERENCES  B. Bollob´as and A. D. Scott, Judicious partitions of bounded-degree graphs, J Graph Theory 46(2) (2004), 131–143.  C. S. Edwards, Some extremal properties of bipartite subgraphs, Canad J Math 25 (1973), 475–485.  C. S. Edwards, An improved lower bound for the number of edges in a largest bipartite subgraph, In: Recent advances in graph theory (Proc.
1 − m m! n n ≥ nm m(m − 1) 1− m! 2n ≥ nm . 2m! Since n ≥ 21, we have m ≥ 3. Hence if (6) holds, then nm < 4m! m m+k , that is, m+k m n