Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Directed graphs without rainbow stars

Daniel Gerbner, Andrzej Grzesik, Cory Palmer, Magdalena Prorok

Abstract

In a rainbow version of the classical Turán problem one considers multiple graphs on a common vertex set, thinking of each graph as edges in a distinct color, and wants to determine the minimum number of edges in each color which guarantees existence of a rainbow copy (having at most one edge from each graph) of a given graph. Here, we prove an optimal solution for this problem for any directed star and any number of colors.

Directed graphs without rainbow stars

Abstract

In a rainbow version of the classical Turán problem one considers multiple graphs on a common vertex set, thinking of each graph as edges in a distinct color, and wants to determine the minimum number of edges in each color which guarantees existence of a rainbow copy (having at most one edge from each graph) of a given graph. Here, we prove an optimal solution for this problem for any directed star and any number of colors.
Paper Structure (4 sections, 6 theorems, 37 equations, 5 figures)

This paper contains 4 sections, 6 theorems, 37 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Rainbow directed $S_{0,q}$
  3. General rainbow directed star
  4. Rainbow directed $S_{1,1}$

Key Result

Theorem 1

For integers $n > c \geq q \geq 1$, every collection of directed graphs $G_1, \ldots, G_c$ on a common set of $n$ vertices containing no rainbow $S_{0,q}$ satisfies Moreover, this bound is sharp.

Figures (5)

  • Figure 1: The optimal construction for 3 colors and forbidden rainbow $S_{0,3}$.
  • Figure 2: The optimal construction for 3 colors and a forbidden rainbow $S_{1,2}$.
  • Figure 3: A construction for 3 colors and a forbidden rainbow $S_{1,2}$ with nonempty sets $A$ and $B$.
  • Figure 4: A construction for 4 colors and a forbidden rainbow $S_{2,2}$ with nonempty sets $A$ and $C$.
  • Figure 5: The optimal construction for at least $4$ colors and a forbidden rainbow $S_{1,1}$.

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 2 more