{"paper":{"title":"An Erd\\H{o}s-Gallai type theorem for vertex colored graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Casey Tompkins, Nika Salia, Oscar Zamora","submitted_at":"2017-12-12T16:54:44Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.04388","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}