## Abstract

It is well-known that the graphs not containing a given graph *H* as a subgraph have bounded chromatic number if and only if *H* is acyclic. Here we consider *ordered graphs*, i.e., graphs with a linear ordering ≺ on their vertex set, and the function

where Forb_{≺}(*H*) denotes the set of all ordered graphs that do not contain a copy of *H*.

If *H* contains a cycle, then as in the case of unordered graphs, *f*_{≺}(*H*)=∞. However, in contrast to the unordered graphs, we describe an infinite family of ordered forests *H* with *f*_{≺}(*H*) =∞. An ordered graph is crossing if there are two edges *uv* and *u*′*v*′ with *u* ≺ *u*′ ≺ *v* ≺ *v*′. For connected crossing ordered graphs *H* we reduce the problem of determining whether *f*_{≺}(*H*) ≠∞ to a family of so-called monotonically alternating trees. For non-crossing *H* we prove that *f*_{≺}(*H*) ≠∞ if and only if *H* is acyclic and does not contain a copy of any of the five special ordered forests on four or five vertices, which we call bonnets. For such forests *H*, we show that *f*_{≺}(*H*)⩽2^{|V(H)|} and that *f*_{≺}(*H*)⩽2|*V* (*H*)|−3 if *H* is connected.

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Axenovich, M., Rollin, J. & Ueckerdt, T. Chromatic Number of Ordered Graphs with Forbidden Ordered Subgraphs.
*Combinatorica* **38, **1021–1043 (2018). https://doi.org/10.1007/s00493-017-3593-0

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### Mathematics Subject Classification (2000)

- 05C15
- 05C35