A graph G is called an odd (even) graph if for every vertex v ∈ V (G), dG(v) is odd (even). Let G be a graph of even order. Scott in 1992 proved that the vertices of every connected graph of even order can be partitioned into some odd induced forests. We denote the minimum number of odd induced subgraphs which partition V (G) by od(G). If all of the subgraphs are forests, then we denote it by odF (G). In this paper, we show that if G is a connected subcubic graph of even order or G is a connected planar graph of even order, then odF (G) ≤ 4. Moreover, we show that for every tree T of even order odF (T ) ≤ 2 and for every unicyclic graph G of even order odF (G) ≤ 3. Also, we prove that if G is claw-free, then V (G) can be partitioned into at most ∆(G)−1 induced forests and possibly one independent set. Furthermore, we demonstrate that the vertex set of the line graph of a tree can be partitioned into at most two odd induced subgraphs and possibly one independent set.
{"title":"Vertex partitioning of graphs into odd induced subgraphs","authors":"A. Aashtab, S. Akbari, M. Ghanbari, Arman Shidani","doi":"10.7151/dmgt.2371","DOIUrl":"https://doi.org/10.7151/dmgt.2371","url":null,"abstract":"A graph G is called an odd (even) graph if for every vertex v ∈ V (G), dG(v) is odd (even). Let G be a graph of even order. Scott in 1992 proved that the vertices of every connected graph of even order can be partitioned into some odd induced forests. We denote the minimum number of odd induced subgraphs which partition V (G) by od(G). If all of the subgraphs are forests, then we denote it by odF (G). In this paper, we show that if G is a connected subcubic graph of even order or G is a connected planar graph of even order, then odF (G) ≤ 4. Moreover, we show that for every tree T of even order odF (T ) ≤ 2 and for every unicyclic graph G of even order odF (G) ≤ 3. Also, we prove that if G is claw-free, then V (G) can be partitioned into at most ∆(G)−1 induced forests and possibly one independent set. Furthermore, we demonstrate that the vertex set of the line graph of a tree can be partitioned into at most two odd induced subgraphs and possibly one independent set.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90190914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Characterization of <i>alpha</i>-excellent <i>2</i>-trees","authors":"Magda Dettlaff, Michael A. Henning, Jerzy Topp","doi":"10.7151/dmgt.2520","DOIUrl":"https://doi.org/10.7151/dmgt.2520","url":null,"abstract":"","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135913896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Domination in graphs and the removal of a matching","authors":"Geoffrey Boyer, Wayne Goddard","doi":"10.7151/dmgt.2516","DOIUrl":"https://doi.org/10.7151/dmgt.2516","url":null,"abstract":"","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135601422","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Disjoint maximal independent sets in graphs and hypergraphs","authors":"P. Dorbec, F. Kacı","doi":"10.7151/dmgt.2500","DOIUrl":"https://doi.org/10.7151/dmgt.2500","url":null,"abstract":"","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71129565","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Rainbow disjoint union of clique and matching in edge-colored complete graph","authors":"Zemin Jin, Junqi Gu","doi":"10.7151/dmgt.2483","DOIUrl":"https://doi.org/10.7151/dmgt.2483","url":null,"abstract":"","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71129848","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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