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

By Kuhn D., Osthus D.

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AN ERDOS-TYPE RESULT AND ITS APPLICATIONS In this section, we will use a classical Erd¨os-type exchange argument in order to prove in a short and unique way some well-known results concerning k-domination and k-dependence. In particular, we will give proves of theorems of Caro and Roditty and of Hopkins and Staton, who already used this principle in a similar way. In this manner, we present a completely self-contained work. Theorem 1. Let G be a graph. If r ≥ 1 is an integer, then there is a partition V (G) = V1 ∪ V2 ∪ · · · ∪ Vr of V (G) such that |N(u) ∩ Vi | ≤ d(u) r (1) for each i ∈ {1, 2, .

Recall that each spanner is connected and thus each spanner in Qn has at least 2 − 1 edges. We do not know whether the bound in (1) can be improved to something like the bound in Theorem 2. In the next section, we obtain the left inequality in (5), then in Section 3 we prove the main part of Theorem 1, the upper bound on k,∞ (n). Finally, in Section 4 we show that the construction of a (2,1)-detour subgraph of [4] with (3 + o(1)) 2n edges is also a 4-detour subgraph in Qn and prove Theorem 3. n 2.

Case 2. w > 2m.

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