A dichotomy for graphs of bounded degeneracy

TL;DR

This paper characterizes when graphs in certain hereditary classes have bounded degeneracy, showing it depends on the presence of specific forbidden subgraphs like complete graphs, bipartite graphs, and forests.

Contribution

It provides a precise dichotomy for bounded degeneracy in hereditary graph classes defined by finite forbidden induced subgraphs.

Findings

Bounded degeneracy occurs iff the forbidden set includes a complete graph, a complete bipartite graph, and a forest.

The result offers a complete classification for these hereditary classes.

It advances understanding of structural properties related to graph degeneracy.

Abstract

We prove that the degeneracy of graphs in a hereditary class defined by a finite set S of forbidden induced subgraphs is bounded if and only if S includes a complete graph, a complete bipartite graph and a forest.