L. K. Jørgensen, Guillermo Pineda-Villavicencio, J. Ugon
{"title":"Linkedness of Cartesian products of complete graphs","authors":"L. K. Jørgensen, Guillermo Pineda-Villavicencio, J. Ugon","doi":"10.26493/1855-3974.2577.25d","DOIUrl":null,"url":null,"abstract":"This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {\\it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. \nWe show that the Cartesian product $K^{d_{1}+1}\\times K^{d_{2}+1}$ of complete graphs $K^{d_{1}+1}$ and $K^{d_{2}+1}$ is $\\floor{(d_{1}+d_{2})/2}$-linked for $d_{1},d_{2}\\ge 2$, and this is best possible. \n%A polytope is said to be {\\it $k$-linked} if its graph is $k$-linked. \nThis result is connected to graphs of simple polytopes. The Cartesian product $K^{d_{1}+1}\\times K^{d_{2}+1}$ is the graph of the Cartesian product $T(d_{1})\\times T(d_{2})$ of a $d_{1}$-dimensional simplex $T(d_{1})$ and a $d_{2}$-dimensional simplex $T(d_{2})$. And the polytope $T(d_{1})\\times T(d_{2})$ is a {\\it simple polytope}, a $(d_{1}+d_{2})$-dimensional polytope in which every vertex is incident to exactly $d_{1}+d_{2}$ edges. \nWhile not every $d$-polytope is $\\floor{d/2}$-linked, it may be conjectured that every simple $d$-polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.","PeriodicalId":8402,"journal":{"name":"Ars Math. Contemp.","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2020-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Ars Math. Contemp.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.26493/1855-3974.2577.25d","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs.
We show that the Cartesian product $K^{d_{1}+1}\times K^{d_{2}+1}$ of complete graphs $K^{d_{1}+1}$ and $K^{d_{2}+1}$ is $\floor{(d_{1}+d_{2})/2}$-linked for $d_{1},d_{2}\ge 2$, and this is best possible.
%A polytope is said to be {\it $k$-linked} if its graph is $k$-linked.
This result is connected to graphs of simple polytopes. The Cartesian product $K^{d_{1}+1}\times K^{d_{2}+1}$ is the graph of the Cartesian product $T(d_{1})\times T(d_{2})$ of a $d_{1}$-dimensional simplex $T(d_{1})$ and a $d_{2}$-dimensional simplex $T(d_{2})$. And the polytope $T(d_{1})\times T(d_{2})$ is a {\it simple polytope}, a $(d_{1}+d_{2})$-dimensional polytope in which every vertex is incident to exactly $d_{1}+d_{2}$ edges.
While not every $d$-polytope is $\floor{d/2}$-linked, it may be conjectured that every simple $d$-polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.