Tur\'an Problems for Mixed Graphs

TL;DR

This paper explores Turán problems in mixed graphs, establishing a Turán density coefficient, an analogue of classical theorems, and revealing that these coefficients can be irrational but always algebraic.

Contribution

It introduces a variational characterization of Turán density for mixed graphs and demonstrates the algebraic nature of these coefficients, highlighting differences from classical extremal numbers.

Findings

Established an analogue of the Erdős-Stone-Simonovits theorem for mixed graphs.

Provided a variational characterization of Turán density coefficients.

Showed Turán density coefficients can be irrational but are always algebraic.

Abstract

We investigate natural Tur\'an problems for mixed graphs, generalizations of graphs where edges can be either directed or undirected. We study a natural \textit{Tur\'an density coefficient} that measures how large a fraction of directed edges an $F$-free mixed graph can have; we establish an analogue of the Erd\H{o}s-Stone-Simonovits theorem and give a variational characterization of the Tur\'an density coefficient of any mixed graph (along with an associated extremal $F$-free family). This characterization enables us to highlight an important divergence between classical extremal numbers and the Tur\'an density coefficient. We show that Tur\'an density coefficients can be irrational, but are always algebraic; for every positive integer $k$, we construct a family of mixed graphs whose Tur\'an density coefficient has algebraic degree $k$.