A $(2,1)$-total labeling of a graph $G$ is an assignment $f$ from the vertex set $V(G)$ and the edge set $E(G)$ to the set $\{0,1,...,k\}$ of nonnegative integers such that $|f(x)-f(y)|\ge 2$ if $x$ is a vertex and $y$ is an edge incident to $x$, and $|f(x)-f(y)|\ge 1$ if $x$ and $y$ are a pair of adjacent vertices or a pair of adjacent edges, for all $x$ and $y$ in $V(G)\cup E(G)$. The $(2,1)$-total labeling number $λ^T_2(G)$ of a graph $G$ is defined as the minimum $k$ among all possible assignments. In [D. Chen and W. Wang. (2,1)-Total labelling of outerplanar graphs. Discr. Appl. Math. 155, 2585--2593 (2007)], Chen and Wang conjectured that all outerplanar graphs $G$ satisfy
$λ^T_2(G) \leq Δ(G)+2$, where $Δ(G)$ is the maximum degree of $G$, while they also showed that it is true for $G$ with $Δ(G)\geq 5$. In this paper, we solve their conjecture completely, by proving that
$λ^T_2(G) \leq Δ(G)+2$ even in the case of $Δ(G)\leq 4 $.
A Tight Meta-theorem for LOCAL Certification of MSO$_2$ Properties within Bounded Treewidth Graphs
Korteweg-de Vries Institute of Mathematics, University of Amsterdam（阿姆斯特丹大学科特维格-德弗里斯数学研究所）；School of Computing, KAIST（韩国科学技术院计算机学院）；CNRS（法国国家科学研究中心）；Institute of Informatics, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw（华沙大学数学、信息学和力学学院信息学研究所）
The independence number of the Birkhoff polytope graph, and applications to maximally recoverable codes
最大可恢复代码是为分布式存储设计的代码，它结合了从单个节点故障中的快速恢复和从灾难性故障中的最佳恢复。Gopalan et al [SODA 2017]研究了网格拓扑中此类码所需的字母表大小，并给出了其组合特征。考虑完全二部图$K_{n,n}$的边的标签来自$F_2^d$，满足以下条件：对于任何简单圈，其边上的标签之和不为零。这是可能的最小d控制在n × n网格拓扑中最大可恢复代码所需的字母表大小。在当前工作之前，已知d在$(\log n)^2$和$n\log n$之间。我们改进了这两个界限，并证明了d在n中是线性的。上界是一个递归构造，它优于随机构造。下界如下首先有关的问题的独立数的Birkhoff多面体图，然后提供严格的界限，它使用的对称群的表示论。
Profile and neighbourhood complexity of graphs excluding a minor and tree-structured graphs
Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS（克莱蒙奥弗涅大学、法国国家科学研究中心、克莱蒙奥弗涅理工学院、圣艾蒂安矿业学院、LIMOS）；Department of Algebra, Faculty of Mathematics and Physics, Charles University（布拉格查理大学数学与物理学院代数系）
