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Turán Problems for Mixed Graphs

Nitya Mani, Edward Yu

Abstract

We investigate natural Turán problems for mixed graphs, generalizations of graphs where edges can be either directed or undirected. We study a natural \textit{Turán density coefficient} that measures how large a fraction of directed edges an $F$-free mixed graph can have; we establish an analogue of the Erdős-Stone-Simonovits theorem and give a variational characterization of the Turán 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 Turán density coefficient. We show that Turán density coefficients can be irrational, but are always algebraic; for every positive integer $k$, we construct a family of mixed graphs whose Turán density coefficient has algebraic degree $k$.

Turán Problems for Mixed Graphs

Abstract

We investigate natural Turán problems for mixed graphs, generalizations of graphs where edges can be either directed or undirected. We study a natural \textit{Turán density coefficient} that measures how large a fraction of directed edges an -free mixed graph can have; we establish an analogue of the Erdős-Stone-Simonovits theorem and give a variational characterization of the Turán density coefficient of any mixed graph (along with an associated extremal -free family). This characterization enables us to highlight an important divergence between classical extremal numbers and the Turán density coefficient. We show that Turán density coefficients can be irrational, but are always algebraic; for every positive integer , we construct a family of mixed graphs whose Turán density coefficient has algebraic degree .
Paper Structure (19 sections, 35 theorems, 86 equations, 22 figures)

This paper contains 19 sections, 35 theorems, 86 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mixed graph fundamentals
  4. Bipartite and undirected mixed graphs
  5. Proof of Theorem 1.6
  6. Uncollapsible mixed graphs
  7. Mixed graphs with one directed edge
  8. General mixed graphs
  9. A variational characterization of θ(F)
  10. A weighted extremal problem
  11. The variational characterization
  12. Algebraic degree of θ(F)
  13. A mixed graph F with irrational θ(F)
  14. θ(F) is always algebraic
  15. Finite families of mixed graphs with arbitrarily high algebraic degree
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\upchi(F)$ denote the chromatic number of $F$, the minimum number of colors required in a proper vertex-coloring of $F$ (where no two adjacent vertices have the same color). Then, we have that

Figures (22)

  • Figure 1: Both $F_1$ and $F_2$ are subgraphs of $G$; $F_1$ is not a subgraph of $F_2$.
  • Figure 2: The first two graphs are examples of mixed graphs (note that a fully undirected graph is still a mixed graph); the last two are not (self-loops and vertices connected by multiple edges are disallowed).
  • Figure 3: A balanced $2$-blowup of a mixed graph
  • Figure 4: Three examples of uncollapsible graphs
  • Figure 5: Head-tail collapsing mixed graph $F$
  • ...and 17 more figures

Theorems & Definitions (111)

  • Theorem 1.1: Erdős-Stone-Simonovits, 1946 erdos_simonovits_1966erdos_stone_1946
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8: Informal
  • Theorem 1.9
  • Theorem 1.10
  • ...and 101 more