A Characterization of Easily Testable Induced Digraphs and $k$-Colored Graphs

TL;DR

This paper characterizes digraphs with polynomial bounds in the induced D-removal lemma and extends the analysis to k-colored complete graphs, including Gallai colorings, providing new polynomial bounds.

Contribution

It completes the characterization of digraphs with polynomial bounds in the induced D-removal lemma and establishes polynomial bounds for Gallai colorings in k-colored graphs.

Findings

Characterization of digraphs with polynomial bounds in the induced D-removal lemma

Polynomial bounds established for Gallai colorings in k-colored graphs

Answer to a question of Alon and Shapira regarding polynomial bounds

Abstract

We complete the characterization of the digraphs $D$ for which the induced $D$-removal lemma has polynomial bounds, answering a question of Alon and Shapira. We also study the analogous problem for $k$-colored complete graphs. In particular, we prove a removal lemma with polynomial bounds for Gallai colorings.