Polynomial bounds for chromatic number. I. Excluding a biclique and an induced tree

TL;DR

This paper proves that graphs excluding a fixed induced tree and a complete bipartite subgraph have degeneracy bounded polynomially in the size of the bipartite graph, advancing understanding of graph coloring constraints.

Contribution

It establishes that for any tree H, the degeneracy of graphs excluding H as an induced subgraph and K_{t,t} is polynomial in t, strengthening previous boundedness results.

Findings

Degeneracy is polynomial in t for graphs excluding a fixed induced tree and K_{t,t}.

Answers an open question on the degeneracy bounds in such graph classes.

Extends prior results on bounded chromatic number and degeneracy in restricted graph classes.

Abstract

Let H be a tree. It was proved by Rodl that graphs that do not contain H as an induced subgraph, and do not contain the complete bipartite graph $K_{t,t}$ as a subgraph, have bounded chromatic number. Kierstead and Penrice strengthened this, showing that such graphs have bounded degeneracy. Here we give a further strengthening, proving that for every tree H, the degeneracy is at most polynomial in t. This answers a question of Bonamy, Pilipczuk, Rzazewski, Thomasse and Walczak.