Pub Date : 2022-04-01DOI: 10.1016/j.disc.2021.112754
Zenghui Fang, Junling Zhou, Lidong Wang
{"title":"Maximum w-cyclic holey group divisible packings and their application to three-dimensional optical orthogonal codes","authors":"Zenghui Fang, Junling Zhou, Lidong Wang","doi":"10.1016/j.disc.2021.112754","DOIUrl":"https://doi.org/10.1016/j.disc.2021.112754","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"13 1","pages":"112754"},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87668718","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 : 2022-04-01DOI: 10.1016/j.disc.2021.112753
Chaohui Chen, Muhuo Liu, Xiaofeng Gu, K. Das
{"title":"Extremal augmented Zagreb index of trees with given numbers of vertices and leaves","authors":"Chaohui Chen, Muhuo Liu, Xiaofeng Gu, K. Das","doi":"10.1016/j.disc.2021.112753","DOIUrl":"https://doi.org/10.1016/j.disc.2021.112753","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"11 1","pages":"112753"},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72709846","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 : 2022-04-01DOI: 10.1016/j.disc.2021.112744
W. Ma, Jinquan Luo
{"title":"Nonexistence of linear codes meeting the Griesmer bound","authors":"W. Ma, Jinquan Luo","doi":"10.1016/j.disc.2021.112744","DOIUrl":"https://doi.org/10.1016/j.disc.2021.112744","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"44 1","pages":"112744"},"PeriodicalIF":0.0,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72844079","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}
J. Balogh, F. Clemen, Bernard Lidick'y, S. Norin, Jan Volec
Denote by $q_n(G)$ the smallest eigenvalue of the signless Laplacian matrix of an $n$-vertex graph $G$. Brandt conjectured in 1997 that for regular triangle-free graphs $q_n(G) leq frac{4n}{25}$. We prove a stronger result: If $G$ is a triangle-free graph then $q_n(G) leq frac{15n}{94}
{"title":"The Spectrum of Triangle-Free Graphs","authors":"J. Balogh, F. Clemen, Bernard Lidick'y, S. Norin, Jan Volec","doi":"10.1137/22m150767x","DOIUrl":"https://doi.org/10.1137/22m150767x","url":null,"abstract":"Denote by $q_n(G)$ the smallest eigenvalue of the signless Laplacian matrix of an $n$-vertex graph $G$. Brandt conjectured in 1997 that for regular triangle-free graphs $q_n(G) leq frac{4n}{25}$. We prove a stronger result: If $G$ is a triangle-free graph then $q_n(G) leq frac{15n}{94}<frac{4n}{25}$. Brandt's conjecture is a subproblem of two famous conjectures of ErdH{o}s: (1) Sparse-Half-Conjecture: Every $n$-vertex triangle-free graph has a subset of vertices of size $lceilfrac{n}{2}rceil$ spanning at most $n^2/50$ edges. (2) Every $n$-vertex triangle-free graph can be made bipartite by removing at most $n^2/25$ edges. In our proof we use linear algebraic methods to upper bound $q_n(G)$ by the ratio between the number of induced paths with 3 and 4 vertices. We give an upper bound on this ratio via the method of flag algebras.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"44 1","pages":"1173-1179"},"PeriodicalIF":0.0,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77250556","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}
Recently, the saturation problem of $0$-$1$ matrices gained a lot of attention. This problem can be regarded as a saturation problem of ordered bipartite graphs. Motivated by this, we initiate the study of the saturation problem of ordered and cyclically ordered graphs. We prove that dichotomy holds also in these two cases, i.e., for a (cyclically) ordered graph its saturation function is either bounded or linear. We also determine the order of magnitude for large classes of (cyclically) ordered graphs, giving infinite many examples exhibiting both possible behaviours, answering a problem of P'alv"olgyi. In particular, in the ordered case we define a natural subclass of ordered matchings, the class of linked matchings, and we start their systematic study, concentrating on linked matchings with at most three links and prove that many of them have bounded saturation function. In both the ordered and cyclically ordered case we also consider the semisaturation problem, where dichotomy holds as well and we can even fully characterize the graphs that have bounded semisaturation function.
{"title":"Saturation of Ordered Graphs","authors":"V. Boskovic, Balázs Keszegh","doi":"10.1137/22m1485735","DOIUrl":"https://doi.org/10.1137/22m1485735","url":null,"abstract":"Recently, the saturation problem of $0$-$1$ matrices gained a lot of attention. This problem can be regarded as a saturation problem of ordered bipartite graphs. Motivated by this, we initiate the study of the saturation problem of ordered and cyclically ordered graphs. We prove that dichotomy holds also in these two cases, i.e., for a (cyclically) ordered graph its saturation function is either bounded or linear. We also determine the order of magnitude for large classes of (cyclically) ordered graphs, giving infinite many examples exhibiting both possible behaviours, answering a problem of P'alv\"olgyi. In particular, in the ordered case we define a natural subclass of ordered matchings, the class of linked matchings, and we start their systematic study, concentrating on linked matchings with at most three links and prove that many of them have bounded saturation function. In both the ordered and cyclically ordered case we also consider the semisaturation problem, where dichotomy holds as well and we can even fully characterize the graphs that have bounded semisaturation function.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"32 1","pages":"1118-1141"},"PeriodicalIF":0.0,"publicationDate":"2022-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85319272","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}
Given a natural $n$, we construct a two-coloring of $mathbb{R}^n$ with the maximum metric satisfying the following. For any finite set of reals $S$ with diameter greater than $5^{n}$ such that the distance between any two consecutive points of $S$ does not exceed one, no isometric copy of $S$ is monochromatic. As a corollary, we prove that any normed space can be two-colored such that all sufficiently long unit arithmetic progressions contain points of both colors.
{"title":"Two-Colorings of Normed Spaces without Long Monochromatic Unit Arithmetic Progressions","authors":"V. Kirova, A. Sagdeev","doi":"10.1137/22M1483700","DOIUrl":"https://doi.org/10.1137/22M1483700","url":null,"abstract":"Given a natural $n$, we construct a two-coloring of $mathbb{R}^n$ with the maximum metric satisfying the following. For any finite set of reals $S$ with diameter greater than $5^{n}$ such that the distance between any two consecutive points of $S$ does not exceed one, no isometric copy of $S$ is monochromatic. As a corollary, we prove that any normed space can be two-colored such that all sufficiently long unit arithmetic progressions contain points of both colors.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"20 1","pages":"718-732"},"PeriodicalIF":0.0,"publicationDate":"2022-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83338635","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 : 2022-03-01DOI: 10.1016/j.disc.2021.112749
Yuanjia Zhang, Xiaoye Liang, J. Koolen
{"title":"The 2-partially distance-regular graphs such that their second largest local eigenvalues are at most one","authors":"Yuanjia Zhang, Xiaoye Liang, J. Koolen","doi":"10.1016/j.disc.2021.112749","DOIUrl":"https://doi.org/10.1016/j.disc.2021.112749","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"4 1","pages":"112749"},"PeriodicalIF":0.0,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75568106","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}
{"title":"Characteristic Dependence of Syzygies of Random Monomial Ideals","authors":"Caitlyn Booms-Peot, D. Erman, Jay Yang","doi":"10.1137/21m1392474","DOIUrl":"https://doi.org/10.1137/21m1392474","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"239 1","pages":"682-701"},"PeriodicalIF":0.0,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76887326","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 : 2022-03-01DOI: 10.1016/j.disc.2021.112740
R. Abel, Thomas Britz, Yudhistira A. Bunjamin, Diana Combe
{"title":"Group divisible designs with block size 4 where the group sizes are congruent to 2 mod 3","authors":"R. Abel, Thomas Britz, Yudhistira A. Bunjamin, Diana Combe","doi":"10.1016/j.disc.2021.112740","DOIUrl":"https://doi.org/10.1016/j.disc.2021.112740","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":"8 1","pages":"112740"},"PeriodicalIF":0.0,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82689748","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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