On edge-ordered graphs with linear extremal functions (2309.10558v2)

Abstract: The systematic study of Tur\'an-type extremal problems for edge-ordered graphs was initiated by Gerbner et al. in 2020. Here we characterize connected edge-ordered graphs with linear extremal functions and show that the extremal function of other connected edge-ordered graphs is $\Omega(n\log n)$. This characterization and dichotomy are similar in spirit to results of F\"uredi et al. (2020) about vertex-ordered and convex geometric graphs. We also extend the study of extremal function of short edge-ordered paths by Gerbner et al. to some longer paths.

• The paper characterizes connected edge-ordered graphs that exhibit linear extremal functions, establishing a dichotomy with graphs showing n log n growth.
• It employs combinatorial techniques and incremental extensions to dissect graph structures while preserving linear extremal behavior.
• The findings enrich the toolkit for addressing Turán-type problems and pave the way for future exploration in combinatorial graph theory.

Characterization of Connected Edge-Ordered Graphs with Linear Extremal Functions

Introduction to Edge-Ordered Graphs and Extremal Functions

Edge-ordered graphs extend the concept of simple graphs by imposing a linear order on the edges. This ordering introduces additional structure, allowing us to paper the interplay between graph theory and order theory. A pivotal concept in this field is the extremal function of a forbidden graph, denoted $_<(n,H)$, which represents the maximum number of edges in an edge-ordered graph on $n$ vertices that does not contain a given edge-ordered graph $H$ as a subgraph. Understanding these functions can reveal deep insights into the combinatorial properties of edge-ordered graphs.

Core Result: A Characterization and Dichotomy

The primary contribution of this paper is a comprehensive characterization of connected edge-ordered graphs that exhibit a linear extremal function. This characterization encapsulates graphs whose extremal functions behave linearly with the size of the graph. Furthermore, the paper establishes a dichotomy: for connected edge-ordered graphs not conforming to this characterization, the extremal function grows at least as fast as $n\log n$.

This dichotomy draws a clear line separating graphs with linear extremal functions from those with faster growth rates, reminiscent of classical results in extremal graph theory yet distinctly tailored to the nuanced context of edge-ordered graphs.

Methodology: Definitions and Theoretical Tools

The exploration of this dichotomy necessitates precise definitions, such as the order chromatic number $\chi_<(G)$, alongside the introduction of edge-ordered bigraphs—a tool for managing the bipartite nature inherent in some of these problems. The methodological backbone of this research hinges on dissecting edge-ordered graphs into simpler structures (semi-caterpillars and extension procedures) and leveraging combinatorial techniques to assess their extremal functions.

One intriguing aspect of the methodology is the use of "extensions" to incrementally build more complex graphs while preserving the property of interest (linear extremal function), thus creating a bridge between minimal examples and a broader class of graphs.

Implications and Future Directions

The results delineated in this paper not only deepen our comprehension of structure within edge-ordered graphs but also open avenues for further investigation. The dichotomy presented here prompts questions about the precise growth rates of extremal functions across different classes of edge-ordered graphs, especially those on the cusp of the linear and super-linear realms.

Furthermore, this research enriches our toolkit for tackling Turán-type problems in edge-ordered graphs, highlighting the potential for cross-pollination between areas such as combinatorial geometry and theoretical computer science. Future work might explore analogous characterizations beyond the confines of connected graphs or explore the territory of almost-linear extremal functions, seeking sharper bounds and refined classifications.

Summary and Contextualization

This paper builds on the pioneering efforts initiated by Gerbner et al. and others, pushing the frontier in our understanding of extremal problems for edge-ordered graphs. By providing a neat characterization of linear extremal functions for a specific class of connected edge-ordered graphs and establishing a dichotomy for the rest, it sets a firm foundation for both theoretical exploration and practical application in the domain of combinatorial graph theory.

In conclusion, these findings underscore the complexity and richness of edge-ordered graphs, inviting mathematicians to explore the intricate interplay between order and connectivity in graph structures.