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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.17055 |
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| _version_ | 1866911166442242048 |
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| author | Li, Binlong Ning, Bo |
| author_facet | Li, Binlong Ning, Bo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17055 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Localized and weighted versions of extremal problems Li, Binlong Ning, Bo Combinatorics 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. |
| title | Localized and weighted versions of extremal problems |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.17055 |