Chromatic Number of Ordered Graphs with Forbidden Ordered Subgraphs
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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.
Mathematics Subject Classification (2000)05C15 05C35
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