Large induced subgraphs with three repeated degrees

Abstract

For any positive integer k, let C(k) denote the least integer such that any n-vertex graph has an induced subgraph with at least n−C(k) vertices, in which at least min⁡{k,n−C(k)} vertices are of the same degree. Caro, Shapira and Yuster initially studied this parameter and showed that Ω(klog⁡k)≤C(k)≤(8k)k. For the first nontrivial case, the authors proved that 3≤C(3)≤6, and the exact value was left as an open problem. In this paper, we first show that 3≤C(3)≤4, improving the former result as well as a recent result of Kogan. For special families of graphs, we prove that C(3)=3 for K5-free graphs, and C(3)=1 for large C2s+1-free graphs. In addition, extending a result of Erdős, Fajtlowicz and Staton, we assert that every Kr-free graph is an induced subgraph of a Kr-free graph in which no degree occurs more than three times.