Pub Date : 2023-01-01DOI: 10.4310/joc.2023.v14.n4.a6
Kalyan Garapaty, D. Lokshtanov, H. K. Maji, A. Pothen
The Erdös-Rényi model is a simple and widely studied model for generating random graphs. Given a positive integer n and a real p between 0 and 1, G(n, p) is the distribution over n-vertex graphs obtained by including, for every unordered pair {u, v} of vertices, the edge uv in the edge set of G independently with probability p. The square of a graph G, denoted by G2, is the graph obtained from G by also adding an edge between every pair of vertices that share at least one common neighbor. A proper k-coloring of a graph G is a function f that assigns to every vertex of G a color f(v) from the set {1, . . . , k} such that no two neighbouring vertices get the same color, and the chromatic number of a graph G is the minimum k so that G has a k-coloring. In a recent article, Cheng, Maji and Pothen [3] consider squares of sparse Erdős-Rényi graphs G(n, p) with p = Θ(1/n) as interesting benchmark instances to evaluate parallel algorithms that color the input graph. These authors prove that if G is sampled from G(n, p) with p = Θ(1/n) then, with high probability, the chromatic number of G2 lies between Ω ( log n log log n ) and O(log n). In this
{"title":"The chromatic number of squares of random graphs","authors":"Kalyan Garapaty, D. Lokshtanov, H. K. Maji, A. Pothen","doi":"10.4310/joc.2023.v14.n4.a6","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n4.a6","url":null,"abstract":"The Erdös-Rényi model is a simple and widely studied model for generating random graphs. Given a positive integer n and a real p between 0 and 1, G(n, p) is the distribution over n-vertex graphs obtained by including, for every unordered pair {u, v} of vertices, the edge uv in the edge set of G independently with probability p. The square of a graph G, denoted by G2, is the graph obtained from G by also adding an edge between every pair of vertices that share at least one common neighbor. A proper k-coloring of a graph G is a function f that assigns to every vertex of G a color f(v) from the set {1, . . . , k} such that no two neighbouring vertices get the same color, and the chromatic number of a graph G is the minimum k so that G has a k-coloring. In a recent article, Cheng, Maji and Pothen [3] consider squares of sparse Erdős-Rényi graphs G(n, p) with p = Θ(1/n) as interesting benchmark instances to evaluate parallel algorithms that color the input graph. These authors prove that if G is sampled from G(n, p) with p = Θ(1/n) then, with high probability, the chromatic number of G2 lies between Ω ( log n log log n ) and O(log n). In this","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"41 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88018899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/joc.2023.v14.n3.a3
Takanori Hida
{"title":"On Mallows’ variation of the Stern–Brocot tree","authors":"Takanori Hida","doi":"10.4310/joc.2023.v14.n3.a3","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n3.a3","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"40 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81370083","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/joc.2023.v14.n4.a3
R. Bennink
{"title":"Counting abelian squares efficiently for a problem in quantum computing","authors":"R. Bennink","doi":"10.4310/joc.2023.v14.n4.a3","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n4.a3","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73450448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.4310/joc.2023.v14.n3.a5
Yasuo Nishii
{"title":"Approximation of Frankl’s conjecture in the complement family","authors":"Yasuo Nishii","doi":"10.4310/joc.2023.v14.n3.a5","DOIUrl":"https://doi.org/10.4310/joc.2023.v14.n3.a5","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"13 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90494685","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.4310/joc.2022.v13.n4.a1
J. V. de Jong
{"title":"Two strong 3-flow theorems for planar graphs","authors":"J. V. de Jong","doi":"10.4310/joc.2022.v13.n4.a1","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n4.a1","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"17 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84420370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.4310/joc.2022.v13.n2.a4
M. Jones
{"title":"A Murnaghan Nakayama rule for schur functions in superspace","authors":"M. Jones","doi":"10.4310/joc.2022.v13.n2.a4","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n2.a4","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84583458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.4310/joc.2022.v13.n3.a1
A. Godbole, Thomas Grubb, K. Han, Bill Kay
{"title":"Threshold progressions in covering and packing contexts","authors":"A. Godbole, Thomas Grubb, K. Han, Bill Kay","doi":"10.4310/joc.2022.v13.n3.a1","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n3.a1","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"60 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83512330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.4310/joc.2022.v13.n1.a2
Stephen J. Young
We provide a lower bound for the spectral radius of the universal cover of irregular graphs in the presence of symmetric edge weights. We use this bound to derive an Alon-Boppana type bound for the second eigenvalue of the normalized Laplacian.
{"title":"The weighted spectrum of the universal cover and an Alon–Boppana result for the normalized Laplacian","authors":"Stephen J. Young","doi":"10.4310/joc.2022.v13.n1.a2","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n1.a2","url":null,"abstract":"We provide a lower bound for the spectral radius of the universal cover of irregular graphs in the presence of symmetric edge weights. We use this bound to derive an Alon-Boppana type bound for the second eigenvalue of the normalized Laplacian.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81861731","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.4310/joc.2022.v13.n4.a2
A. Sanitt, John M. Talbot
{"title":"The Turan density problem for hypergraphs","authors":"A. Sanitt, John M. Talbot","doi":"10.4310/joc.2022.v13.n4.a2","DOIUrl":"https://doi.org/10.4310/joc.2022.v13.n4.a2","url":null,"abstract":"","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"35 1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83548073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-12DOI: 10.4310/JOC.2023.v14.n4.a2
N.Ph. Deniskin, M. Benzi
Subgraph centrality, introduced by Estrada and Rodr'iguez-Vel'azquez in [12], has become a widely used centrality measure in the analysis of networks, with applications in biology, neuroscience, economics and many other fields. It is also worthy of study from a strictly mathematical point of view, in view of its connections to topics in spectral graph theory, number theory, analytic matrix functions, and combinatorics. In this paper we present some new results and a list of open questions about subgraph centrality and other node centrality measures based on graph walks.
{"title":"New results and open problems on subgraph centrality","authors":"N.Ph. Deniskin, M. Benzi","doi":"10.4310/JOC.2023.v14.n4.a2","DOIUrl":"https://doi.org/10.4310/JOC.2023.v14.n4.a2","url":null,"abstract":"Subgraph centrality, introduced by Estrada and Rodr'iguez-Vel'azquez in [12], has become a widely used centrality measure in the analysis of networks, with applications in biology, neuroscience, economics and many other fields. It is also worthy of study from a strictly mathematical point of view, in view of its connections to topics in spectral graph theory, number theory, analytic matrix functions, and combinatorics. In this paper we present some new results and a list of open questions about subgraph centrality and other node centrality measures based on graph walks.","PeriodicalId":44683,"journal":{"name":"Journal of Combinatorics","volume":"32 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75618021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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