On edge-ordered graphs with linear extremal functions

TL;DR

This paper characterizes connected edge-ordered graphs with linear extremal functions and shows a dichotomy in their extremal growth, extending previous work on paths and related graph classes.

Contribution

It provides a complete characterization of connected edge-ordered graphs with linear extremal functions and establishes a dichotomy in their extremal behavior.

Findings

Connected edge-ordered graphs with linear extremal functions are characterized.

Other connected edge-ordered graphs have extremal functions of at least (n  log n).

Extended the study of extremal functions to longer edge-ordered paths.

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.