About chromatic uniqueness of some complete tripartite graphs

Approaches Which Output Infinitely Many Graphs With Small Local Antimagic Chromatic Number

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we (i) give a sufficient condition for a graph with one pendant to have $χ_{la}\ge 3$. A necessary and s…

同态与VC维阈值：谱与分离

Homomorphism and VC-dimension thresholds: spectra and separations

该研究揭示同态与VC维阈值具无限谱且非单调，证明四类全局简单性阈值两两不同，确定多部图VC维阈值及同态阈值的下界与聚点。

H-chromatic symmetric functions

We introduce $H$-chromatic symmetric functions, $X_{G}^{H}$, which use the $H$-coloring of a graph $G$ to define a generalization of Stanley's chromatic symmetric functions. We say two graphs $G_1$ and $G_2$ are $H$-chromatically equivalent if $X_{G_1}^{H} = X_{G_2}^{H}$, and use this idea to study uniqueness results for $H$-chromatic symmetric functions, with a particular emphasis on the case $H$ is a complete bipartite graph. We also show that several of the classical bases of the space of symmetric functions, i.e. the monomial symmetric functions, power sum symmetric functions, and elementa…

多部图的局部反幻色数

Local antimagic chromatic number of partite graphs

本文研究由局部反幻边标号诱导的正常顶点着色，确定了完全三部图 K_{1,m,n} 及奇偶性不同的 K_{m,n} 的 r 个副本的局部反幻色数。

三部染色临界图的边谱过饱和

Edge-spectral supersaturation for tripartite color-critical graphs

该研究针对色数为3的两类染色临界图，证明超过紧谱Turán阈值会迫使出现最优多项式数量的对应图副本，扩展了边谱存在性定理至三色过饱和结果。

On local antimagic chromatic number of lexicographic product graphs

Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic if $G$ admits a local antimagic labeling. A bijection $f : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $f^+(u) \ne f^+(v)$, where $f^+(u) = \sum_{e\in E(u)} f(e)$, and $E(u)$ is the set of edges incident to $u$. Thus, any local antimagic labeling induces a proper vertex coloring of $G$ if vertex $v$ is assigned the color $f^+(v)$. The local antimagic chromatic number, denoted $χ_{la}(G)$, is the minimum number…

构造无限多个具有局部反幻色数3的偶数大小图的算法方法

An algorithmic approach in constructing infinitely many even size graphs with local antimagic chromatic number 3

本文提出一种算法方法，构造了无限多个偶数大小非正则三部图族，并证明这些图及其不同阶和大小分支的变体均具有局部反幻色数3。

关于移位图中顶点临界诱导子图的注记

A note on vertex-critical induced subgraphs of shift graphs

本文研究了移位图中顶点临界诱导子图的性质，发现每个k色移位图包含唯一的k-顶点临界子图，并给出了其显式描述。

完全多重图的Schur正性分类

A Schur-positivity classification for complete multipartite graphs

本文通过证明完全多重图K_λ在Schur基下非负展开当且仅当λ_i∈{1,2}或λ=(3,2^β)时，给出了完全多重图的Schur正性分类，扩展了二部图和三部图的正性分类结果。

Interval graph limits

We work out the graph limit theory for dense interval graphs. The theory developed departs from the usual description of a graph limit as a symmetric function $W(x,y)$ on the unit square, with $x$ and $y$ uniform on the interval $(0,1)$. Instead, we fix a $W$ and change the underlying distribution of the coordinates $x$ and $y$. We find choices such that our limits are continuous. Connections to random interval graphs are given, including some examples. We also show a continuity result for the chromatic number and clique number of interval graphs. Some results on uniqueness of the limit descri…