Construction of a family of graphs with a small induced proper subgraph with minimum degree 3

Abstract

We investigate the following question proposed by Erdős: Is there a constant c such that, for each n, if G is a graph with n vertices, 2 n - 1 edges, and δ ( G ) ⩾ 3 , then G contains an induced proper subgraph H with at least cn vertices and δ ( H ) ⩾ 3 ? Previously we showed that there exists no such constant c by constructing a family of graphs whose induced proper subgraph with minimum degree 3 contains at most ⌈ n ⌉ vertices. In this paper we present a construction of a family of graphs whose largest induced proper subgraph with minimum degree 3 is K 4 . Also a similar construction of a graph with n vertices and α n + β edges is given.