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

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.