Duality and $χ^<$-Boundedness of Ordered Graphs

TL;DR

This work analyzes ordered graphs under edge- and order-preserving homomorphisms, delivering a complete singleton duality classification and a robust theory of $\chi^<$-boundedness tailored to order. It introduces the monotone-matching framework and greedy ordered coloring to obtain efficient colorings, and proves a Sparse Incomparability Lemma for ordered homomorphisms. The results yield a dense homomorphism order with explicit gaps, and reveal that, unlike unordered graphs, ordered graphs exhibit a richer gap structure tied to consecutive monotone matchings and isolated edges. Overall, the paper provides foundational duality, bounding, and density results that deepen understanding of ordered graph homomorphisms and their structural implications.

Abstract

We show that there exists only one duality pair for ordered graphs. We will also define a corresponding definition of $χ^<$-boundedness for ordered graphs and show that all ordered graphs are $χ^<$-bounded and prove an analogy of Gyárfás-Sumner conjecture for ordered graphs. We also prove an analogy of Sparse Incomparability Lemma for ordered graphs. We then use this result to show classes of ordered graphs that form a dense order under ordered homomorphisms. We also show that compared to graphs, ordered graphs have more gaps, defined by consecutive monotone matchings and by even more generic pairs of ordered graphs differing by one isolated edge.

Figures (3)

• Figure 1: Ordered Homomorphism $f$ and Independent Intervals.
• Figure 2: $M_n, M^{LR}_n, M^{RL}_n$ and $M^{+}_n$
• Figure 3: Sparse Incomparability Lemma Mapping

Theorems & Definitions (32)

• Definition 3.1
• Theorem 3.1
• Proposition 4.1
• proof
• proof
• proof
• Lemma 4.4
• proof
• Theorem 5.1
• proof