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Graphs without a rainbow path of length 3

Sebastian Babiński, Andrzej Grzesik

Abstract

In 1959 Erdős and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. Here we study a rainbow version of their theorem, in which one considers $k \geq 1$ graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximum number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any $k \geq 1$.

Graphs without a rainbow path of length 3

Abstract

In 1959 Erdős and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. Here we study a rainbow version of their theorem, in which one considers graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximum number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any .
Paper Structure (6 sections, 2 theorems, 54 equations, 2 figures)

This paper contains 6 sections, 2 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General claims
  3. Three and four colors
  4. Five and six colors
  5. At least seven colors
  6. Conclusion

Key Result

Theorem 1

For every $\varepsilon > 0$ there exists $n_0 \in \mathbb{N}$ such that for every $n \geq n_0$, $k \geq 1$ and graphs $G_1, G_2, \ldots, G_k$ on a common set of $n$ vertices, each graph having at least $(f(k) + \varepsilon)\frac{n^2}{2}$ edges, where there exists a rainbow path with 3 edges. Moreover, the above bound on the number of edges is asymptotically optimal for each $k \geq 1$.

Figures (2)

  • Figure 1: A clustered graph for 3 colors.
  • Figure 2: Two possible types of extremal constructions for $k=5$.

Theorems & Definitions (36)

  • Theorem 1
  • Definition 2
  • Theorem 3
  • Claim 4
  • proof
  • Claim 5
  • proof
  • Claim 6
  • proof
  • Claim 7
  • ...and 26 more