Graphs without a rainbow path of length 3

TL;DR

This paper extends Erdős and Gallai's classical theorem to a rainbow setting, determining the maximum edges in multiple graphs on the same vertices without creating a rainbow path of length three.

Contribution

It establishes the asymptotically optimal bound for the maximum edges in k graphs avoiding a rainbow path of length three, generalizing a classical result.

Findings

Proves the asymptotic bound for k graphs without rainbow P3.

Extends classical extremal graph theory to a rainbow setting.

Provides a foundation for further rainbow path investigations.

Abstract

In 1959 Erd\H{o}s 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 $k \geq 1$.