Localized and weighted versions of extremal problems

Abstract

Malec and Tompkins (EUJC, 2023) considered the localized versions of Turán-type problems, and proved a localized theorem on Erdős-Gallai Theorem on paths. Zhao and Zhang (JGT, 2025) gave a long proof of a localized version of Erdős-Gallai Theorem on cycles. In this paper, we consider several types of generalization of Turán-type problems, that is, localized versions, weighted versions, and generalized Turán-type problems, and their connectedness. We first present very short proofs for recent results of Malec-Tompkins and Zhao-Zhang, respectively. We use Small Path Double Cover Conjecture, which was proposed by Bondy (JGT, 1990) and confirmed by Hao Li (JGT, 1990), to prove a weighted localized Turán-type theorem on paths. We prove localized versions of Balister-Bollobás-Riordan-Schelp Theorem (JCTB, 2003) on paths and Erdős-Gallai Theorem on matchings, respectively. We show that our first localized result implies Balister-Bollobás- Riordan-Schelp Theorem, Erdős-Gallai Theorem, and Malec-Tompkins Theorem on paths. Finally, we present generalized Turán-style generalizations of the Malec-Tompkin's Theorem, and discuss the relationship between some previous theorems in different motivations.

16 pages