An Erd\H{o}s-Gallai type theorem for vertex colored graphs

TL;DR

This paper extends the classical Erd ext{"o}s-Gallai theorem to vertex-colored graphs, establishing sharp bounds on edges for paths with endpoints of different colors, and generalizes to trees and related conjectures.

Contribution

It proves that the original Erd ext{"o}s-Gallai bound applies to colored graphs and introduces a generalized problem for trees, advancing understanding of path and tree structures in colored graphs.

Findings

Erd ext{"o}s-Gallai bound of $kn$ applies to colored graphs with no certain paths.

Introduces a colored version of the Erd ext{"o}s-Gallai theorem.

Proposes a generalization related to the Erd ext{"o}s-Sós conjecture.

Abstract

While investigating odd-cycle free hypergraphs, Gy\H{o}ri and Lemons introduced a colored version of the classical theorem of Erd\H{o}s and Gallai on $P_k$-free graphs. They proved that any graph $G$ with a proper vertex coloring and no path of length $2k+1$ with endpoints of different colors has at most $2kn$ edges. We show that Erd\H{o}s and Gallai's original sharp upper bound of $kn$ holds for their problem as well. We also introduce a version of this problem for trees and present a generalization of the Erd\H{o}s-S\'os conjecture.