Large induced subgraphs with equated maximum degree

AbstractFor a graph G, denote by fk(G) the smallest number of vertices that must be deleted from G so that the remaining induced subgraph has its maximum degree shared by at least k vertices. It is not difficult to prove that there are graphs for which already f2(G)≥n(1−o(1)), where n is the number of vertices of G. It is conjectured that fk(G)=Θ(n) for every fixed k. We prove this for k=2,3. While the proof for the case k=2 is easy, already the proof for the case k=3 is considerably more difficult. The case k=4 remains open.A related parameter, sk(G), denotes the maximum integer m so that there are k vertex-disjoint subgraphs of G, each with m vertices, and with the same maximum degree. We prove that for every fixed k, sk(G)≥n/k−o(n). The proof relies on probabilistic arguments