. We establish a Hanani-Tutte style characterization for hierarchical partial planarity 3 and initiate the study of partitioned partial
。我们建立了层次偏平面度3的Hanani-Tutte风格表征，并开始了分区偏度的研究
{"title":"Hanani-Tutte and Hierarchical Partial Planarity","authors":"M. Schaefer","doi":"10.1137/21m1464749","DOIUrl":"https://doi.org/10.1137/21m1464749","url":null,"abstract":". We establish a Hanani-Tutte style characterization for hierarchical partial planarity 3 and initiate the study of partitioned partial","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72858350","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":"The Lower Bound Theorem for $d$-Polytopes with $2{d}+1$ Vertices","authors":"Guillermo Pineda-Villavicencio, D. Yost","doi":"10.1137/21m144832x","DOIUrl":"https://doi.org/10.1137/21m144832x","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83192911","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 : 2022-11-11DOI: 10.48550/arXiv.2211.05930
Yan Cao, Guangming Jing, Rong Luo, V. Mkrtchyan, Cun-Quan Zhang, Yue Zhao
Mkrtchyan and Steffen [J. Graph Theory, 70 (4), 473--482, 2012] showed that every class II simple graph can be decomposed into a maximum $Delta$-edge-colorable subgraph and a matching. They further conjectured that every graph $G$ with chromatic index $Delta(G)+k$ ($kgeq 1$) can be decomposed into a maximum $Delta(G)$-edge-colorable subgraph (not necessarily class I) and a $k$-edge-colorable subgraph. In this paper, we first generalize their result to multigraphs and show that every multigraph $G$ with multiplicity $mu$ can be decomposed into a maximum $Delta(G)$-edge-colorable subgraph and a subgraph with maximum degree at most $mu$. Then we prove that every graph $G$ with chromatic index $Delta(G)+k$ can be decomposed into two class I subgraphs $H_1$ and $H_2$ such that $Delta(H_1) = Delta(G)$ and $Delta(H_2) = k$, which is a variation of their conjecture.
{"title":"Decomposition of class II graphs into two class I graphs","authors":"Yan Cao, Guangming Jing, Rong Luo, V. Mkrtchyan, Cun-Quan Zhang, Yue Zhao","doi":"10.48550/arXiv.2211.05930","DOIUrl":"https://doi.org/10.48550/arXiv.2211.05930","url":null,"abstract":"Mkrtchyan and Steffen [J. Graph Theory, 70 (4), 473--482, 2012] showed that every class II simple graph can be decomposed into a maximum $Delta$-edge-colorable subgraph and a matching. They further conjectured that every graph $G$ with chromatic index $Delta(G)+k$ ($kgeq 1$) can be decomposed into a maximum $Delta(G)$-edge-colorable subgraph (not necessarily class I) and a $k$-edge-colorable subgraph. In this paper, we first generalize their result to multigraphs and show that every multigraph $G$ with multiplicity $mu$ can be decomposed into a maximum $Delta(G)$-edge-colorable subgraph and a subgraph with maximum degree at most $mu$. Then we prove that every graph $G$ with chromatic index $Delta(G)+k$ can be decomposed into two class I subgraphs $H_1$ and $H_2$ such that $Delta(H_1) = Delta(G)$ and $Delta(H_2) = k$, which is a variation of their conjecture.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90804151","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 : 2022-10-20DOI: 10.48550/arXiv.2210.11184
I. Mogilnykh, K. Vorob'ev
We prove that any completely regular code with minimum eigenvalue in any geometric graph G corresponds to a completely regular code in the clique graph of G. Studying the interrelation of these codes, a complete characterization of the completely regular codes in the Johnson graphs J(n,w) with covering radius w-1 and strength 1 is obtained. In particular this result finishes a characterization of the completely regular codes in the Johnson graphs J(n,3). We also classify the completely regular codes of strength 1 in the Johnson graphs J(n,4) with only one case for the eigenvalues left open.
{"title":"On completely regular codes with minimum eigenvalue in geometric graphs","authors":"I. Mogilnykh, K. Vorob'ev","doi":"10.48550/arXiv.2210.11184","DOIUrl":"https://doi.org/10.48550/arXiv.2210.11184","url":null,"abstract":"We prove that any completely regular code with minimum eigenvalue in any geometric graph G corresponds to a completely regular code in the clique graph of G. Studying the interrelation of these codes, a complete characterization of the completely regular codes in the Johnson graphs J(n,w) with covering radius w-1 and strength 1 is obtained. In particular this result finishes a characterization of the completely regular codes in the Johnson graphs J(n,3). We also classify the completely regular codes of strength 1 in the Johnson graphs J(n,4) with only one case for the eigenvalues left open.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76777784","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}
M. R. Pournaki, M. Poursoltani, N. Terai, S. Yassemi
{"title":"Simplicial Complexes Satisfying Serre's Condition versus the Ones Which Are Cohen-Macaulay in a Fixed Codimension","authors":"M. R. Pournaki, M. Poursoltani, N. Terai, S. Yassemi","doi":"10.1137/21m1439687","DOIUrl":"https://doi.org/10.1137/21m1439687","url":null,"abstract":"","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81949896","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 : 2022-10-11DOI: 10.48550/arXiv.2210.05194
Ziling Heng, Xinran Wang
In ``Infinite families of near MDS codes holding $t$-designs, IEEE Trans. Inform. Theory, 2020, 66(9), pp. 5419-5428'', Ding and Tang made a breakthrough in constructing the first two infinite families of NMDS codes holding $2$-designs or $3$-designs. Up to now, there are only a few known infinite families of NMDS codes holding $t$-designs in the literature. The objective of this paper is to construct new infinite families of NMDS codes holding $t$-designs. We determine the weight enumerators of the NMDS codes and prove that the NMDS codes hold $2$-designs or $3$-designs. Compared with known $t$-designs from NMDS codes, ours have different parameters. Besides, several infinite families of optimal locally recoverable codes are also derived via the NMDS codes.
{"title":"New infinite families of near MDS codes holding $t$-designs and optimal locally recoverable codes","authors":"Ziling Heng, Xinran Wang","doi":"10.48550/arXiv.2210.05194","DOIUrl":"https://doi.org/10.48550/arXiv.2210.05194","url":null,"abstract":"In ``Infinite families of near MDS codes holding $t$-designs, IEEE Trans. Inform. Theory, 2020, 66(9), pp. 5419-5428'', Ding and Tang made a breakthrough in constructing the first two infinite families of NMDS codes holding $2$-designs or $3$-designs. Up to now, there are only a few known infinite families of NMDS codes holding $t$-designs in the literature. The objective of this paper is to construct new infinite families of NMDS codes holding $t$-designs. We determine the weight enumerators of the NMDS codes and prove that the NMDS codes hold $2$-designs or $3$-designs. Compared with known $t$-designs from NMDS codes, ours have different parameters. Besides, several infinite families of optimal locally recoverable codes are also derived via the NMDS codes.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80369540","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}
. We consider the following augmentation problem: Given a rigid graph G = ( V,E ), 3 find a minimum cardinality edge set F such that the graph G (cid:48) = ( V,E ∪ F ) is globally rigid. We 4 provide a min-max theorem and a polynomial-time algorithm for this problem for several types of 5 rigidity, such as rigidity in the plane or on the cylinder. Rigidity is often characterized by some 6 sparsity properties of the underlying graph and global rigidity is characterized by redundant rigidity 7 (where the graph remains rigid after deleting an arbitrary edge) and 2-or 3-vertex-connectivity. 8 Hence, to solve the above-mentioned problem, we define and solve polynomially a combinatorial 9 optimization problem family based on these sparsity and connectivity properties. This family also 10 includes the problem of augmenting a k -tree-connected graph to a highly k -tree-connected and 2-11 connected graph. Moreover, as an interesting consequence, we give an optimal solution to the 12 so-called global rigidity pinning problem, where we aim to find a minimum cardinality vertex set X 13 for a rigid graph G = ( V,E ), such that the graph G + K X is globally rigid in R 2 where K X denotes 14 the complete graph on the vertex set X .
{"title":"Globally Rigid Augmentation of Rigid Graphs","authors":"C. Király, András Mihálykó","doi":"10.1137/21m1432417","DOIUrl":"https://doi.org/10.1137/21m1432417","url":null,"abstract":". We consider the following augmentation problem: Given a rigid graph G = ( V,E ), 3 find a minimum cardinality edge set F such that the graph G (cid:48) = ( V,E ∪ F ) is globally rigid. We 4 provide a min-max theorem and a polynomial-time algorithm for this problem for several types of 5 rigidity, such as rigidity in the plane or on the cylinder. Rigidity is often characterized by some 6 sparsity properties of the underlying graph and global rigidity is characterized by redundant rigidity 7 (where the graph remains rigid after deleting an arbitrary edge) and 2-or 3-vertex-connectivity. 8 Hence, to solve the above-mentioned problem, we define and solve polynomially a combinatorial 9 optimization problem family based on these sparsity and connectivity properties. This family also 10 includes the problem of augmenting a k -tree-connected graph to a highly k -tree-connected and 2-11 connected graph. Moreover, as an interesting consequence, we give an optimal solution to the 12 so-called global rigidity pinning problem, where we aim to find a minimum cardinality vertex set X 13 for a rigid graph G = ( V,E ), such that the graph G + K X is globally rigid in R 2 where K X denotes 14 the complete graph on the vertex set X .","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90781249","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 : 2022-09-15DOI: 10.5817/cz.muni.eurocomb23-071
Yan Gu, Yiting Jiang, D. Wood, Xuding Zhu
Assume $lambda={k_1,k_2, ldots, k_q}$ is a partition of $k_{lambda} = sum_{i=1}^q k_i$. A $lambda$-list assignment of $G$ is a $k_lambda$-list assignment $L$ of $G$ such that the colour set $bigcup_{v in V(G)}L(v)$ can be partitioned into $lambda= q$ sets $C_1,C_2,ldots,C_q$ such that for each $i$ and each vertex $v$ of $G$, $L(v) cap C_i ge k_i$. We say $G$ is emph{$lambda$-choosable} if $G$ is $L$-colourable for any $lambda$-list assignment $L$ of $G$. The concept of $lambda$-choosability is a refinement of choosability that puts $k$-choosability and $k$-colourability in the same framework. If $lambda$ is close to $k_lambda$, then $lambda$-choosability is close to $k_lambda$-colourability; if $lambda$ is close to $1$, then $lambda$-choosability is close to $k_lambda$-choosability. This paper studies Hadwiger‘s Conjecture in the context of $lambda$-choosability. Hadwiger‘s Conjecture is equivalent to saying that every $K_t$-minor-free graph is ${1 star (t-1)}$-choosable for any positive integer $t$. We prove that for $t ge 5$, for any partition $lambda$ of $t-1$ other than ${1 star (t-1)}$, there is a $K_t$-minor-free graph $G$ that is not $lambda$-choosable. We then construct several types of $K_t$-minor-free graphs that are not $lambda$-choosable, where $k_lambda - (t-1)$ gets larger as $k_lambda-lambda$ gets larger.
假设$lambda={k_1,k_2, ldots, k_q}$是$k_{lambda} = sum_{i=1}^q k_i$的一个分区。$G$的$lambda$ -list赋值是$k_lambda$ -list赋值$L$的$G$，这样颜色集$bigcup_{v in V(G)}L(v)$可以被划分为$lambda= q$集合$C_1,C_2,ldots,C_q$，这样对于$i$和$G$的每个顶点$v$, $L(v) cap C_i ge k_i$。我们说$G$是emph{$lambda$-可选择}的，如果$G$是$L$ -可着色的，对于$G$的任何$lambda$ -list赋值$L$。$lambda$ -可选择性的概念是对可选择性的改进，将$k$ -可选择性和$k$ -可着色性放在同一个框架中。如果$lambda$接近$k_lambda$，那么$lambda$ -可选择性接近$k_lambda$ -可着色性;如果$lambda$接近$1$，那么$lambda$ -choosability接近$k_lambda$ -choosability。本文在$lambda$ -可选择性的背景下研究哈德维格猜想。哈维格猜想等价于说，对于任何正整数$t$，每个$K_t$ -无次元图都是${1 star (t-1)}$ -可选的。我们证明了对于$t ge 5$，对于除${1 star (t-1)}$以外的$t-1$的任何分区$lambda$，存在一个不能$lambda$选择的无$K_t$次元图$G$。然后，我们构造了几种类型的$K_t$ -minor-free图形，这些图形不能选择$lambda$ -，其中$k_lambda - (t-1)$随着$k_lambda-lambda$变大而变大。
{"title":"Refined List Version of Hadwiger's Conjecture","authors":"Yan Gu, Yiting Jiang, D. Wood, Xuding Zhu","doi":"10.5817/cz.muni.eurocomb23-071","DOIUrl":"https://doi.org/10.5817/cz.muni.eurocomb23-071","url":null,"abstract":"Assume $lambda={k_1,k_2, ldots, k_q}$ is a partition of $k_{lambda} = sum_{i=1}^q k_i$. A $lambda$-list assignment of $G$ is a $k_lambda$-list assignment $L$ of $G$ such that the colour set $bigcup_{v in V(G)}L(v)$ can be partitioned into $lambda= q$ sets $C_1,C_2,ldots,C_q$ such that for each $i$ and each vertex $v$ of $G$, $L(v) cap C_i ge k_i$. We say $G$ is emph{$lambda$-choosable} if $G$ is $L$-colourable for any $lambda$-list assignment $L$ of $G$. The concept of $lambda$-choosability is a refinement of choosability that puts $k$-choosability and $k$-colourability in the same framework. If $lambda$ is close to $k_lambda$, then $lambda$-choosability is close to $k_lambda$-colourability; if $lambda$ is close to $1$, then $lambda$-choosability is close to $k_lambda$-choosability. This paper studies Hadwiger‘s Conjecture in the context of $lambda$-choosability. Hadwiger‘s Conjecture is equivalent to saying that every $K_t$-minor-free graph is ${1 star (t-1)}$-choosable for any positive integer $t$. We prove that for $t ge 5$, for any partition $lambda$ of $t-1$ other than ${1 star (t-1)}$, there is a $K_t$-minor-free graph $G$ that is not $lambda$-choosable. We then construct several types of $K_t$-minor-free graphs that are not $lambda$-choosable, where $k_lambda - (t-1)$ gets larger as $k_lambda-lambda$ gets larger.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83341469","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}
The natural matroid of an integer polymatroid was introduced to show that a simple construction of integer polymatroids from matroids yields all integer polymatroids. As we illustrate, the natural matroid can shed much more light on integer polymatroids. We focus on characterizations of integer polymatroids using their bases, their circuits, and their cyclic flats along with the rank of each cyclic flat and each element; we offer some new characterizations and insights into known characterizations.
{"title":"The Natural Matroid of an Integer Polymatroid","authors":"Joseph E. Bonin, C. Chun, Tara Fife","doi":"10.1137/22m1521122","DOIUrl":"https://doi.org/10.1137/22m1521122","url":null,"abstract":"The natural matroid of an integer polymatroid was introduced to show that a simple construction of integer polymatroids from matroids yields all integer polymatroids. As we illustrate, the natural matroid can shed much more light on integer polymatroids. We focus on characterizations of integer polymatroids using their bases, their circuits, and their cyclic flats along with the rank of each cyclic flat and each element; we offer some new characterizations and insights into known characterizations.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73492313","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}
If $mathcal{C}$ is a minor-closed class of matroids, the class $mathcal{C}'$ of integer polymatroids whose natural matroids are in $mathcal{C}$ is also minor closed, as is the class $mathcal{C}'_k$ of $k$-polymatroids in $mathcal{C}'$. We find the excluded minors for $mathcal{C}'_2$ when $mathcal{C}$ is (i) the class of binary matroids, (ii) the class of matroids with no $M(K_4)$-minor, and, combining those, (iii) the class of matroids whose connected components are cycle matroids of series-parallel networks. In each case the class $mathcal{C}$ has finitely many excluded minors, but that is true of $mathcal{C}'_2$ only in case (ii). We also introduce the $k$-natural matroid, a variant of the natural matroid for a $k$-polymatroid, and use it to prove that these classes of 2-polymatroids are closed under 2-duality.
{"title":"The Excluded Minors for Three Classes of 2-Polymatroids Having Special Types of Natural Matroids","authors":"Joseph E. Bonin, Kevin Long","doi":"10.1137/22m1521134","DOIUrl":"https://doi.org/10.1137/22m1521134","url":null,"abstract":"If $mathcal{C}$ is a minor-closed class of matroids, the class $mathcal{C}'$ of integer polymatroids whose natural matroids are in $mathcal{C}$ is also minor closed, as is the class $mathcal{C}'_k$ of $k$-polymatroids in $mathcal{C}'$. We find the excluded minors for $mathcal{C}'_2$ when $mathcal{C}$ is (i) the class of binary matroids, (ii) the class of matroids with no $M(K_4)$-minor, and, combining those, (iii) the class of matroids whose connected components are cycle matroids of series-parallel networks. In each case the class $mathcal{C}$ has finitely many excluded minors, but that is true of $mathcal{C}'_2$ only in case (ii). We also introduce the $k$-natural matroid, a variant of the natural matroid for a $k$-polymatroid, and use it to prove that these classes of 2-polymatroids are closed under 2-duality.","PeriodicalId":21749,"journal":{"name":"SIAM J. Discret. Math.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84531266","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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