A characterization of edge-ordered graphs with almost linear extremal functions

Abstract: The systematic study of Tur\'an-type extremal problems for edge-ordered graphs was initiated by Gerbner et al. arXiv:2001.00849. They conjectured that the extremal functions of edge-ordered forests of order chromatic number 2 are $n^{1+o(1)}$. Here we resolve this conjecture proving the stronger upper bound of $n2^{O(\sqrt{\log n})}$. This represents a gap in the family of possible extremal functions as other forbidden edge-ordered graphs have extremal functions $\Omega(n^c)$ for some $c>1$. However, our result is probably not the last word: here we conjecture that the even stronger upper bound of $n\log^{O(1)}n$ also holds for the same set of extremal functions.

• The paper establishes that edge-ordered forests with order chromatic number 2 have extremal functions bounded by n·2^(O(√(log n))).
• The density increment argument is applied recursively to construct subgraphs that maintain a nearly constant average degree despite a reduced edge count.
• The findings reveal sharper distinctions in extremal functions between edge-ordered and simple graphs, opening avenues for further research on forbidden substructures.

A New Upper Bound for the Extremal Functions of Edge-Ordered Forests

Introduction

Turán-type problems have been a central theme in combinatorial mathematics, focusing on the maximal size a structure can achieve while avoiding a particular substructure. This paper addresses such problems within the framework of edge-ordered graphs, expanding our understanding of their extremal functions. Specifically, it addresses a conjecture by Gerbner et al. regarding edge-ordered forests with order chromatic number 2, providing a stronger upper bound on their extremal functions.

Main Results

The paper successfully proves a conjectured upper bound for the extremal functions of edge-ordered forests of order chromatic number 2, establishing that $_<(n,H)=n\cdot2^{O(\sqrt{\log n})}$ for such graphs. This result narrows down the family of possible extremal functions for edge-ordered graphs, introducing a distinction from previously understood behaviors in these and related graph families.

One noteworthy aspect of this result is the gap it identifies in the extremal functions of edge-ordered graphs, a gap smaller than known gaps for simple graphs but significant enough to mark a divergence in behavior between these and other graph families.

Methodology

The proof employs a density increment argument, a common strategy in extremal graph theory, tailored to the characteristics of edge-ordered graphs. It centers around a key theorem that, given an edge-ordered graph avoiding a particular edge-ordered forest, constructs a subgraph with substantially fewer edges but only a constant factor loss in average degree. This construction is repeated recursively, leading to the main result.

Significantly, the recursive application of the density increment technique is carefully calibrated, taking into consideration the unique edge ordering conditions. The selection of subgraphs and application of grids for partitioning the edge set are vital components of the proof.

Implications and Future Directions

The findings of this paper have both theoretical and practical implications for the study of edge-ordered graphs and Turán-type extremal problems. Theoretically, the paper contributes to a deeper understanding of the behavior of edge-ordered graphs with respect to forbidden substructures. Practically, the results may inform algorithms and applications where such structures are relevant, including in areas such as network theory and data structure optimization.

The paper also sets the stage for future research, particularly in addressing the conjecture of edge-ordered forests of order chromatic number 2 having extremal functions of $n\log^{O(1)}n$. Furthermore, it opens questions regarding the existence of edge-ordered forests with extremal functions exceeding $\Omega(n\log n)$, posing a challenge that mirrors historical developments in the analysis of forbidden structures.

Conclusion

In solving a conjectured upper bound for the extremal functions of certain edge-ordered forests, this paper furthers our understanding of the Turán-type problems for edge-ordered graphs. Through a meticulous density increment argument, it showcases the nuanced behavior of these graphs under avoidance conditions. Future explorations in this area, motivated by the conjectures and questions raised, promise to reveal further complexities in the behavior of edge-ordered graphs and other combinatorial structures.