, has small maximum degree, a small edge-hitting set can be constructed from a small vertex-hitting set. On the other hand, a big maximum degree forces a large packing of ? r -models

. Lemma-7, If f r is the vertex-Erd? os-Pósa gap of M(? r ), then the edge-Erd? os-Pósa gap of M(? r ) is less than 2kr · f r (k)

, be an integer and let f r is the vertex-Erd? os-Pósa gap of M(? r ) We want to prove that if G contains less than k edge-disjoint models of ? r , then it has a ? r -edge-hitting set of size less than 2kr · f r (k) According to Remark 1, we can assume that G is biconnected. If it is not the case, we consider its biconnected components separately (if it has no biconnected component then the lemma is trivial)

?. First, Notice that if G does not contain k edge-disjoint ? r -models, it does not contain k vertexdisjoint ? r -models either. Consequently, there is a set X ? V(G) meeting every ? r model of G and such that |X| f r (k) Let us consider the set Y ? E(G) of edges incident to vertices of X, i.e. Y = {{u, v} ? E(G), u ? X}. Remark that as ?(G) < 2kr, we have |Y | 2kr · f r (k). Now, assume that there is a ? r -model in G not having edges in Y. None of its vertices is in X, which is contradictory

, Corollary 2. An edge-gap of O(k 3 r 3 ) for M(? r ) can be derived from Proposition

, Proof of Theorem 1. It follows from the application of Lemma 7 to the estimations of the vertex-Erd? os-Pósa gap of ? r given in Corollary 1

, The main question is whether for every planar graph J, the class M(J) satisfies this edge variant of the Erd? os-Pósa property. As for the vertex version, it is easy to see that the planarity of J is necessary. For instance, if J = K 5 , consider as graph G an n-vertex toroidal wall, which is a 3-regular graph embeddable in the torus that contains K 5 as a minor. One can check that G does not contain two edge-disjoint models of K 5

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