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

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.

💡 Research Summary

The paper investigates three fundamental aspects of ordered graphs—homomorphism dualities, χ<‑boundedness, and order density—providing a comprehensive theory that parallels, yet simplifies, many classical results for unordered graphs. An ordered graph is defined as a triple (V,E,≤) where ≤ is a total order on the vertex set, and an ordered homomorphism f : V→V′ must preserve both adjacency and the vertex order.

The first major contribution is a complete classification of singleton homomorphism dualities for ordered graphs. The authors introduce the notion of an ordered core (a minimal subgraph that admits no homomorphism to a proper ordered subgraph) and prove that every ordered core is either a monotone matching Mk or a totally ordered complete graph Kk. Moreover, the pair (Mk, Kk) is shown to be the unique singleton duality: for any ordered graph G, Mk→G if and only if G↛Kk, and conversely G↛Kk if and only if G→Kk. The proof hinges on the invariant λ(G), the maximum number of pairwise non‑intersecting edges, which satisfies λ(Mk)=k and λ(Kk)=k−1. By applying the greedy coloring algorithm to G, the authors construct a minimal image Ag that contains a directed path P_g; the size of this path forces g−1<k, establishing the duality.

The second contribution adapts the classical χ‑boundedness concept to ordered graphs. Instead of the clique number ω(G), the authors use η(G), the size of the largest monotone matching contained in G. A family of ordered graphs is χ<‑bounded if there exists a function f such that χ<(G)≤f(η(G)) for every member G, where χ<(G) is the minimum number of independent intervals needed to partition V(G). Theorem 6.1 proves that all ordered graphs satisfy χ<(G)≤2·η(G)+1. The argument uses the greedy interval‑coloring algorithm, which is shown to be optimal (Proposition 4.1) and runs in linear time O(|V|+|E|). This result implies that the ordered chromatic number is always bounded by a linear function of the monotone matching size, establishing universal χ<‑boundedness.

The third major result is an ordered‑graph analogue of the Sparse Incomparability Lemma, which the authors employ to study the homomorphism order among ordered graphs. They demonstrate that, when restricting to graphs whose connected components have at least three vertices, the homomorphism order is dense: for any two graphs G<H (i.e., G→H and H↛G) there exists a third graph F with G→F→H unless the pair forms a “gap.” The paper identifies two natural sources of gaps: consecutive monotone matchings (Mk and Mk+1) and pairs that differ by a single isolated edge. These gaps are more abundant than in the unordered setting, reflecting the richer structure introduced by the vertex order.

Technical highlights include: (i) a clean linear‑time greedy algorithm that computes χ<(G) exactly, (ii) the introduction of λ(G) as an ordered analogue of the clique number, and (iii) the use of monotone matchings as the fundamental obstruction for homomorphisms. The authors also discuss potential extensions, such as dualities for restricted classes of ordered graphs, but leave these for future work.

Overall, the paper provides a unified framework for ordered graph homomorphisms, showing that the added order constraint leads to a surprisingly simple duality theory, universal χ<‑boundedness, and a densely ordered homomorphism lattice punctuated by well‑characterized gaps. These insights bridge graph theory, model theory, and algorithm design, and they open new avenues for applying ordered graph concepts in areas ranging from Ramsey theory to computational problems involving ordered data structures.