Summary: Graphs and Combinatorics 3, 203-211 (1987)
© Springer-Verlag 1987
N. Alon 1., J. Kahn 2.* and P.D. Seymour 3
1 Department of Mathematics, Tel Aviv University, Tel Aviv, Israel and
Bell Communications Research, Morristown, NJ 07960, USA
2 Department of Mathematics and Center for OR, Rutgers University, New Brunswick,
NJ 08903, USA
3 BellCommunications Research, Morristown, NJ 07960, USA
Abstract. A graph H is d-degenerateifevery subgraph ofit contains a vertex ofdegree smaller than
d. For a graph G, let ad(G)denote the maximum number of vertices of an induced d-degenerate
subgraph of G. Sharp lowers bounds for %(G)in terms of the degree sequence of G are obtained,
and the minimum number of edges of a graph G with n vertices and ~2(G)< m is determined
precisely for all m < n.
All graphs considered here are finite and simple. A graph H is d-degenerateif
every non-null subgraph of it contains a vertex of degree smaller than d. Thus
1-degenerate graphs are graphs with no edges and 2-degenerate graphs are forests.