Chromatic Number of Ordered Graphs with Forbidden Ordered Subgraphs

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

$${f_ \prec }\left( H \right) = \sup \left\{ {\chi \left( G \right)|G \in For{b_ \prec }\left( H \right)} \right\},$$

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 uv′ with uu′ ≺ vv′. 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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Correspondence to Maria Axenovich.

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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