Turán problems for Edgeordered graphs
Abstract
In this paper we initiate a systematic study of the Turán problem for edgeordered graphs. A simple graph is called $\textit{edgeordered}$, if its edges are linearly ordered. An isomorphism between edgeordered graphs must respect the edgeorder. A subgraph of an edgeordered graph is itself an edgeordered graph with the induced edgeorder. We say that an edgeordered graph $G$ $\textit{avoids}$ another edgeordered graph $H$, if no subgraph of $G$ is isomorphic to $H$. The $\textit{Turán number}$ of an edgeordered graph $H$ is the maximum number of edges in an edgeordered graph on $n$ vertices that avoids $H$. We study this problem in general, and establish an ErdősStoneSimonovitstype theorem for edgeordered graphs  we discover that the relevant parameter for the Turán number of an edgeordered graph is its $\textit{order chromatic number}$. We establish several important properties of this parameter. We also study Turán numbers of edgeordered paths, star forests and the cycle of length four. We make strong connections to DavenportSchinzel theory, the theory of forbidden submatrices, and show an application in Discrete Geometry.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.00849
 Bibcode:
 2020arXiv200100849G
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 41 pages. Added missing references and updates parts of Section 2