Duality and $\chi^<$-Boundedness of Ordered Graphs

TL;DR

This paper explores the duality and boundedness properties of ordered graphs, establishing unique duality pairs, proving all are $ ext{chi}^<$-bounded, and extending key conjectures and lemmas to the ordered graph context.

Contribution

It introduces the concept of $ ext{chi}^<$-boundedness for ordered graphs, proves all ordered graphs are $ ext{chi}^<$-bounded, and extends fundamental conjectures and lemmas to ordered graphs.

Findings

Existence of a unique duality pair for ordered graphs

All ordered graphs are $ ext{chi}^<$-bounded

Analogies of Gyárfás-Sumner conjecture and Sparse Incomparability Lemma established

Abstract

We show that there exists only one duality pair for ordered graphs. We will also define a corresponding definition of $\chi^<$-boundedness for ordered graphs and show that all ordered graphs are $\chi^<$-bounded and prove an analogy of Gy\'arf\'as-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.