Large induced subgraphs with equated maximum degree

For 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) = ( 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 vertexdisjoint 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.

• Type: Journal
• Year: 2010
• Where: DM
• Authors: Yair Caro, Raphael Yuster
• Keywords: DM 2010, Maximum Degree, Probabilistic Arguments, Smallest Number