Twin-width and generalized coloring numbers

TL;DR

This paper establishes bounds on admissibility and coloring numbers for graphs with bounded twin-width and no large complete bipartite subgraph, demonstrating exponential dependencies and constructing graphs that achieve these bounds.

Contribution

It introduces bounds on coloring parameters for graphs with bounded twin-width and no $K_{s,s}$, and constructs graphs that realize these bounds.

Findings

Bounds on $r$-admissibility and $r$-coloring numbers are exponential in $r$ for such graphs.

Existence of graphs that attain these exponential bounds.

Graphs with no $K_{s,s}$ and bounded twin-width exhibit controlled coloring properties.

Abstract

In this paper, we prove that a graph $G$ with no $K_{s,s}$-subgraph and twin-width $d$ has $r$-admissibility and $r$-coloring numbers bounded from above by an exponential function of $r$ and that we can construct graphs achieving such a dependency in $r$.