Chromatic number of ordered graphs with forbidden ordered subgraphs

TL;DR

This paper investigates the chromatic number bounds of ordered graphs avoiding certain subgraphs, revealing new infinite families and conditions for finiteness based on the structure and crossing properties of the forbidden subgraphs.

Contribution

It characterizes when the chromatic number is bounded for ordered graphs avoiding specific subgraphs, introducing new families and conditions distinct from unordered graph cases.

Findings

Infinite family of ordered forests with infinite chromatic bounds

Finite bounds for non-crossing forests avoiding 'bonnets'

Bounds of 2^|V(H)| and 2|V(H)|-3 for certain forests

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(H) = sup{chi(G) | G in Forb(H)} 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) is infinity. However, in contrast to the unordered graphs, we describe an infinite family of ordered forests H with infinite 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) is finite to a family of so-called monotonically alternating trees. For non-crossing H we prove that f(H) is finite 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.