<p>In this paper we continue the study of a natural generalization of Turán's forbidden subgraph problem and the Ruzsa–Szemerédi problem. Let <span></span><math> <semantics> <mrow> <mrow> <msub> <mtext>ex</mtext> <mi>F</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>,</mo> <mi>G</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mrow> <annotation> ${text{ex}}_{F}(n,G)$</annotation> </semantics></math> denote the maximum number of edge-disjoint copies of a fixed simple graph <span></span><math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> <annotation> $F$</annotation> </semantics></math> that can be placed on an <span></span><math> <semantics> <mrow> <mrow> <mi>n</mi> </mrow> </mrow> <annotation> $n$</annotation> </semantics></math>-vertex ground set without forming a subgraph <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> whose edges are from different <span></span><math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> <annotation> $F$</annotation> </semantics></math>-copies. The case when both <span></span><math> <semantics> <mrow> <mrow> <mi>F</mi> </mrow> </mrow> <annotation> $F$</annotation> </semantics></math> and <span></span><math> <semantics> <mrow> <mrow> <mi>G</mi> </mrow> </mrow> <annotation> $G$</annotation> </semantics></math> are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when <span></span><math> <semantics> <mrow>
{"title":"Multicolor Turán numbers II: A generalization of the Ruzsa–Szemerédi theorem and new results on cliques and odd cycles","authors":"Benedek Kovács, Zoltán Lóránt Nagy","doi":"10.1002/jgt.23147","DOIUrl":"10.1002/jgt.23147","url":null,"abstract":"<p>In this paper we continue the study of a natural generalization of Turán's forbidden subgraph problem and the Ruzsa–Szemerédi problem. Let <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mtext>ex</mtext>\u0000 \u0000 <mi>F</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>G</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> ${text{ex}}_{F}(n,G)$</annotation>\u0000 </semantics></math> denote the maximum number of edge-disjoint copies of a fixed simple graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math> that can be placed on an <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $n$</annotation>\u0000 </semantics></math>-vertex ground set without forming a subgraph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> whose edges are from different <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math>-copies. The case when both <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>F</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $F$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> are triangles essentially gives back the theorem of Ruzsa and Szemerédi. We extend their results to the case when <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141649705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Partial cubes are graphs that can be isometrically embedded into hypercubes. Convex cycles play an important role in the study of partial cubes. In this paper, we prove that a regular partial cube is a hypercube (resp., a Doubled Odd graph, an even cycle of length where ) if and only if all its convex cycles are 4-cycles (resp., 6-cycles, -cycles). In particular, the partial cubes whose all convex cycles are 4-cycles are equivalent to almost-median graphs. Therefore, we conclude that regular almost-median graphs are exactly hypercubes, which generalizes the result by Mulder—regular median graphs are hypercubes.
{"title":"A characterization of regular partial cubes whose all convex cycles have the same lengths","authors":"Yan-Ting Xie, Yong-De Feng, Shou-Jun Xu","doi":"10.1002/jgt.23126","DOIUrl":"10.1002/jgt.23126","url":null,"abstract":"<p>Partial cubes are graphs that can be isometrically embedded into hypercubes. Convex cycles play an important role in the study of partial cubes. In this paper, we prove that a regular partial cube is a hypercube (resp., a Doubled Odd graph, an even cycle of length <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $2n$</annotation>\u0000 </semantics></math> where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mo>⩾</mo>\u0000 \u0000 <mn>4</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $ngeqslant 4$</annotation>\u0000 </semantics></math>) if and only if all its convex cycles are 4-cycles (resp., 6-cycles, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mi>n</mi>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $2n$</annotation>\u0000 </semantics></math>-cycles). In particular, the partial cubes whose all convex cycles are 4-cycles are equivalent to almost-median graphs. Therefore, we conclude that regular almost-median graphs are exactly hypercubes, which generalizes the result by Mulder—regular median graphs are hypercubes.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141504322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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