Brian Alspach
Honeycomb toroidal graphs are trivalent Cayley graphs on generalized dihedral groups. We examine the two historical threads leading to these graphs, some of the properties that have been established, and some open problems.
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On The 2-Spanning Cyclability Of Honeycomb Toroidal Graphs
本文研究蜂窝环面图的2-生成环可容性，探讨图中任意两个不同顶点能否分属由2-因子构成的两个不同环。
Symmetries of the Honeycomb toroidal graphs
本文针对蜂窝环面图这一参数化三正则 Cayley 图族，确定了每个图的完整自同构群，从而解决 Alspach 综述中提出的两个研究问题。
We show that the cop number of toroidal graphs is at most 3. This resolves a conjecture by Schroeder from 2001 which is implicit in a question by Andreae from 1986.
The {\em perfect matching complex} of a graph is the simplicial complex on the edge set of the graph with facets corresponding to perfect matchings of the graph. This paper studies the perfect matching complexes, $\mathcal{M}_p(H_{k \times m\times n})$, of honeycomb graphs. For $k = 1$, $\mathcal{M}_p(H_{1\times m\times n})$ is contractible unless $n\ge m=2$, in which case it is homotopy equivalent to the $(n-1)$-sphere. Also, $\mathcal{M}_p(H_{2\times 2\times 2})$ is homotopy equivalent to the wedge of two 3-spheres. The proofs use discrete Morse theory.
It's well known that every planar graph is $4$-colorable. A toroidal graph is a graph that can be embedded on a torus. It's proved that every toroidal graph is $7$-colorable. A proper coloring of a graph is called \emph{odd} if every non-isolated vertex has at least one color that appears an odd number of times in its neighborhood. The smallest number of colors that admits an odd coloring of a graph $ G $ is denoted by $χ_{o}(G)$. In this paper, we prove that if $G$ is tortoidal, then $χ_{o}\left({G}\right)\le9$; Note that $K_7$ is a toroidal graph, the upper bound is no less than $7$.
Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a graph $G$ is denoted by $χ_o(G)$. We say $G$ is odd $k$-colorable if $χ_o(G)\le k$. This notion is introduced very recently by Petruševski and Škrekovski, who proved that if $G$ is planar then $ χ_{o}(G) \leq 9 $. A toroidal graph is a graph that can be embedded on a torus. Note that a $K_7$ is a toroidal graph, $χ_{o}(G)\leq7$. Tian and Yin proved that every …
We investigate the toroidal expanse of an embedded graph G, that is, the size of the largest toroidal grid contained in G as a minor. In the course of this work we introduce a new embedding density parameter, the stretch of an embedded graph G, and use it to bound the toroidal expanse from above and from below within a constant factor depending only on the genus and the maximum degree. We also show that these parameters are tightly related to the planar crossing number of G. As a consequence of our bounds, we derive an efficient constant factor approximation algorithm for the toroidal expanse …
Maximum run length in a toroidal grid graph
环形网格图是圈的笛卡儿积，网格图中哈密顿圈的游程长度定义为使得任意r条连续边在任意维数上包含的边不超过一条的最大个数r.通过构造性的方法，我们对几类格点图中的哈密顿圈的最大可能游程给出了界。
Odd coloring is a proper coloring with an additional restriction that every non-isolated vertex has some color that appears an odd number of times in its neighborhood. The minimum number of colors $k$ that can ensure an odd coloring of a graph $G$ is denoted by $χ_o(G)$. We say $G$ is $k$-colorable if $χ_o(G)\le k$. This notion is introduced very recently by Petruševski and Škrekovski, who proved that if $G$ is planar then $ χ_{o}(G) \leq 9 $. A toroidal graph is a graph that can be embedded on a torus. Note that a $K_7$ is a toroidal graph, $χ_{o}(G)\leq7$. In this paper, we proved that, ever…
Vertex Arboricity of Toroidal Graphs with a Forbidden Cycle
图$G$的顶点荫度$a(G)$是最小$k$，使得$V(G)$可以被划分为$k$集合，其中每个集合诱导一个森林。对于平面图$G$，已知$a(G)\leq 3$ .在最近的两篇论文中，证明了对于某些$k\in\{3, 4, 5, 6, 7\}$，不含$k$ -圈的平面图的点荫度至多为2。对于环形图$G$，已知$a(G)\leq 4$ .让我们考虑下面的问题：没有$k$ -圈的环形图的点荫度是否至多为2？众所周知，这个问题对于k=3是成立的，最近，Zhang证明了这个问题对于$k=5$也是成立的。由于一个5点完全图是一个没有任何$k$ -圈的环面图，并且$k\geq 6$的顶点荫度至少为3，唯一未知的情况是k=4。我们解决了这种情况下，肯定的，即，我们表明，没有4圈的环形图的顶点荫度最多为2。
Toroidal graphs containing neither $K_5^{-}$ nor 6-cycles are 4-choosable
图$G$的可选性$χ_\ell(G)$是最小$k$，使得在每个顶点处具有可用的$k$颜色保证适当的着色。给定一个环形图$G$，已知有$χ_\ell(G)\leq 7$和$χ_\ell(G)=7$当且仅当$G$包含$K_7$。Cai，Wang和Zhu证明了不含7-圈的环形图$G$是6-可选的，并且$χ_\ell(G)=6$当且仅当$G$包含$K_6$ .证明了不含6-圈的环形图$G$是5-可选的，并推测了$χ_\ell(G)=5$当且仅当$G$包含$K_5$ .我们通过构造一个无限族的非4-可着色的环面图来反驳这个猜想，其中既没有$K_5$，也没有长度至少为6的圈;而且，这个图族是可嵌入除了平面和射影平面之外的任何表面上的。相反，我们证明了以下稍弱的陈述建议朱：环形图既不包含$K^-_5$（一个$K_5$失踪的一个边缘），也不是6-圈是4-choosable。这是尖锐的，因为禁止两个结构中的一个并不能确保图是4-可选的。
We give an exact characterization of 3-colorability of triangle-free graphs drawn in the torus, in the form of 186 "templates" (graphs with certain faces filled by arbitrary quadrangulations) such that a graph from this class is not 3-colorable if and only if it contains a subgraph matching one of the templates. As a consequence, we show every triangle-free graph drawn in the torus with edge-width at least six is 3-colorable, a key property used in an efficient 3-colorability algorithm for triangle-free toroidal graphs.
A Note on (3,1)-Choosable Toroidal Graphs
$(L,d)^*$着色是一种映射$ϕ$，它将颜色$ϕ(v)\in L(v)$分配给每个顶点$v\in V(G)$，使得$v$的至多$d$邻居接收颜色$ϕ(v)$ .一个图称为$(m,d)^*$ -可选择的，如果$G$允许对每个列表分配$L$都有$(L,d)^*$ -着色，且对所有$v\in V(G)$都有$|L(v)|\geq m$.本文证明了对于某些$l \in \{5,7\}$，不含邻接三角形且不含6-圈和$l$ -圈的环面图是$(3,1)^*$ -可选的.
Toroidal Embeddings of Right Groups
本文研究曲面上右群的Cayley图的嵌入。刻画了一类右群的特征，即右群有一个环面但无平面Cayley图，使得右群的生成系有一个极小生成系作为因子.
The theory of Dvorak, Kral, and Thomas (2015) shows that a 4-critical triangle-free graph embedded in the torus has only a bounded number of faces of length greater than 4 and that the size of these faces is also bounded. We study the natural reduction in such embedded graphs - identification of opposite vertices in 4-faces. We give a computer-assisted argument showing that there are exactly four 4-critical triangle-free irreducible toroidal graphs in which this reduction cannot be applied without creating a triangle. Using this result, we show that every 4-critical triangle-free graph embedde…
Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The annihilator graph of $R$ is defined as the undirected graph $AG(R)$ with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $ann_R(xy)\neq ann_R(x)\cup ann_R(y)$. In this paper, all rings whose annihilator graphs can be embed on the plane or torus are classified.
The resonance graphs of nanotubes and toroidal polyhexes
本文针对一般纳米管和环面多六边形，通过加强割集流和梯子条件，给出了两个完美匹配位于同一共振图连通分量的有效判据，并利用环面上简单环的同伦类简化了交替六边形情形。
Toroidal graphs without $K_{5}^{-}$ and 6-cycles
本文针对不含$K_{5}^{-}$和6-环的环面图给出结构刻画，借此强化Choi的结论，证明其弱退化度至多3、DP相关色数及Alon-Tarsi数至多4，且结果具有紧性。
Full support of the Kasteleyn operator associated with a bipartite toroidal graph
嵌入在环面上的二部图中的完美匹配定义了图的面上的高度函数和$\Z^2$中的关联高度变化向量。这些匹配通过二元Laurent多项式的四个评估的组合来枚举，称为Kasteleyn算子，其二次系数（i，j）是直到符号，具有高度变化的完美匹配的数量（i，j）。因此，Kasteleyn算子的牛顿多边形是高度变化向量的凸包。在这篇文章中，我们证明了任何点的整数坐标在该多边形是实现一个完美的匹配。
Criticality of counterexamples to toroidal edge-hamiltonicity
Grünbaum和Nash-Williams的一个著名猜想提出4-连通环形图是哈密尔顿图。4-连通平面图和射影-可平面图的相应结果分别由Tutte和Thomas和Yu证明，他们使用归纳论证证明了一个更强的结果，即每条边都在Hamilton圈上。然而，这个更强的性质并不适用于4-连通环形图：Dahassen构造了反例。因此，标准的归纳方法将不适用于环面。一种可能的方法来修改它是通过表征的情况下，一些边缘是不是一个汉密尔顿循环。我们提供了一个贡献，在这个方向上，通过显示，明显的概括，在一定意义上是至关重要的Mrassen的反例。
A Kempe swap in a proper coloring interchanges the colors on some maximal connected 2-colored subgraph. Two $k$-colorings are $k$-equivalent if we can transform one into the other using Kempe swaps. The triangulated toroidal grid, $T[m\times n]$, is formed from (a toroidal embedding of) the Cartesian product of $C_m$ and $C_n$ by adding parallel diagonals inside all 4-faces. Mohar and Salas showed that not all 4-colorings of $T[m\times n]$ are 4-equivalent. In contrast, Bonamy, Bousquet, Feghali, and Johnson showed that all 6-colorings of $T[m\times n]$ are 6-equivalent. They asked whether the…
The obstructions for toroidal graphs with no $K_{3,3}$'s
环形图的禁止子图和子图有很多。我们考虑没有$K_{3,3}$ -子图的环面图与没有$K_{3,3}$ -子图的环面图重合。这些图承认一个独特的分解成平面组件，并有简短的障碍物列表。给出了不含$K_{3,3}$的环面图的4个禁止子图和11个禁止子图的完整列表，并证明了这些列表是充分的.
Decomposition of toroidal graphs without some subgraphs
本文提出统一方法，证明不含特定圈（如3、4、6、7圈）的环面图的公共超类均为(2,1)-可分解，推广了Lu和Li的结果。
A longest sequence $S$ of distinct vertices of a graph $G$ such that each vertex of $S$ dominates some vertex that is not dominated by its preceding vertices, is called a Grundy dominating sequence; the length of $S$ is the Grundy domination number of $G$. In this paper we study the Grundy domination number in the four standard graph products: the Cartesian, the lexicographic, the direct, and the strong product. For each of the products we present a lower bound for the Grundy domination number which turns out to be exact for the lexicographic product and is conjectured to be exact for the stro…
Planar-Toroidal Decomposition of $K_{12}$
研究证明K12无法分解为一个平面图和一个扭面图，并揭示了平面图与扭面图边数的关系。
We investigate multiple variants of the game Cops and Robbers. Playing it on an $n \times n$ toroidal chess graph, the game is varied by defining moves for cops and robbers differently, always mimicking moves of certain chess pieces. In these cases, the cop number is completely determined.
Counting unlabelled toroidal graphs with no K33-subdivisions
我们提供了一个描述的未标记的枚举技术，完整的证明，可以通过取代2-极点网络的核心图的边缘规范获得的图形。利用不含K33-剖分的环面图和射影-可平面图的结构定理，我们利用这些技巧得到了它们的无标号计数。
K5 and K3,3 are Toroidal Penny Graphs
本文探讨了便士图与平环面上球体最优填充的联系，并证明了$K5$和$K_{3,3}$是平正方环面上的便士图。
Characterization and enumeration of toroidal K_{3,3}-subdivision-free graphs
我们描述了2-连通非平面环形图的结构，没有K_{3,3} -细分，使用适当的替代平面网络到某些图形的边缘称为环形核心。结构结果是基于[A. Gagarin和W. Kocay，“包含K_5-子划分的嵌入图”，Ars Combin. 64（2002），33-49]。它允许在线性时间内识别这些图，并且可以通过使用类似于[A]的方法来枚举不包含K_{3,3} -细分并且具有最小顶点度2或3的标记2-连通环形图。Gagarin，G. Labelle和P. Leroux，“Counting labeled可平面图without a K_{3,3} -subdivision”，提交，arXiv：math.CO/0406140，（2004）]。
Commutative rings with toroidal zero-divisor graphs
设$R$是交换环，$Γ(R)$表示交换环的零因子图.本文研究了可嵌入$Γ(R)$的紧致Riemann曲面的亏格数，并给出了所有有限交换环$R$（直到同构），使得$Γ(R)$是环面或平面的.
A {\em total coloring} of a graph $G$ is an assignment of colors to the vertices and the edges of $G$ such that every pair of adjacent/incident elements receive distinct colors. The {\em total chromatic number} of a graph $G$, denoted by $\chiup''(G)$, is the minimum number of colors in a total coloring of $G$. The well-known Total Coloring Conjecture (TCC) says that every graph with maximum degree $Δ$ admits a total coloring with at most $Δ+ 2$ colors. A graph is {\em $1$-toroidal} if it can be drawn in torus such that every edge crosses at most one other edge. In this paper, we investigate t…
The Alon-Tarsi number $AT(G)$ of a graph $G$ is the smallest $k$ for which there is an orientation $D$ of $G$ with max indegree $k-1$ such that the number of even and odd circulations contained in D are different. In this paper, we show that the Alon--Tarsi number of toroidal grids $T_{m,n}=C_m\Box C_n$ equals $4$ when $m,n$ are both odd and $3$ otherwise.
Toroidal involutory Cayley graphs
本文研究含单位元有限交换环的对合Cayley图，对所有该图为环面图的环R完成了分类，丰富了Cayley图的拓扑嵌入相关结果。
DP-coloring, also known as correspondence coloring, is introduced by Dvo{ř}{á}k and Postle. It is a generalization of list coloring. In this paper, we show that every connected toroidal graph without triangles adjacent to $5$-cycles has minimum degree at most three unless it is a 2-connected $4$-regular graph with Euler characteristic $ε(G) = 0$. Consequently, every toroidal graph without triangles adjacent to $5$-cycles is DP-$4$-colorable. In the final, we show that every planar graph without two certain subgraphs is DP-$4$-colorable. As immediate consequences, (i) every planar graph without…
Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $Γ(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $Γ(G)$ if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of $K_{3,3}$, $K_{1,5}$, $C_6$, $P_5$, or $P_6$ as a subgraph.
Towards Characterization of 5-List-Colorability of Toroidal Graphs
本文通过计算机辅助枚举列出了满足特定性质的环面图5-可选性极小障碍，验证了不存在额外障碍，支持了环面图5-可选性与5-可着色性等价的猜想。
In this paper, we study visibility representations of graphs that are embedded on a torus or a Klein bottle. Mohar and Rosenstiehl showed that any toroidal graph has a visibility representation on a flat torus bounded by a parallelogram, but left open the question whether one can assume a rectangular flat torus, i.e., a flat torus bounded by a rectangle. Independently the same question was asked by Tamassia and Tollis. We answer this question in the positive. With the same technique, we can also show that any graph embedded on a Klein bottle has a visibility representation on the rectangular f…
Embeddability of hieroglyphs in the torus and the intersection form
本文研究单顶点带图的环面嵌入问题，给出其可嵌入环面的判定准则，从而解决 Skopenkov 提出的相关问题。
Toroidal maps : Schnyder woods, orthogonal surfaces and straight-line representations
施奈德木是满足简单局部性质的平面地图边缘的方向和着色。我们提出了一个推广的施奈德伍兹嵌入在环面上的图形与应用程序的图形绘制。我们证明了这个新的对象上的几个属性。其中，我们证明了嵌入在环面上的图允许这样的Schnyder木材当且仅当它是一个本质上3-连通的环面映射。我们证明，这些施奈德树林可以用于将本质上3连通的环形映射的泛覆盖嵌入到无限且周期性的正交曲面上。最后，我们使用这种嵌入，以获得一个直线平坦的环面表示的任何环形地图在一个多项式大小的网格。
We present enumerations of a class of toroidal graphs which give rise to semi-equivelar maps. There are eleven different types of semi-equivelar maps on the torus. These are of the types $\{3^{6}\}$, $\{4^{4}\}$, $\{6^{3}\}$, $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $\{4, 8^{2}\}$, $\{3, 12^{2}\}$, $\{4, 6, 12\}$, $\{3, 4, 6, 4\}$. We know the classification of the maps of types $\{3^{6}\}$, $\{4^{4}\}$, $\{6^{3}\}$ on the torus. In this article, we attempt to classify maps of types $\{3^{3}, 4^{2}\}$, $\{3^{2}, 4, 3, 4\}$, $\{3, 6, 3, 6\}$, $\{3^{4}, 6\}$, $…
Toroidal embeddings of cubic projective plane obstructions
本文对六个三次射影平面障碍图在环面中的嵌入进行了等价分类，完善了 Glover 和 Huneke 关于三次图射影平面嵌入障碍的结果。
Spectral Theory of the Toroidal 3D Queen Graph
本文研究了三维环形后妃图G_n的邻接谱，利用傅里叶特征将其对角化，推导出每个频率a对应的特征值λ(a)=nμ(a)-13，其中μ(a)计数与a正交的后妃方向数，且在n为奇数时给出特征值的乘法公式。
Improper coloring of toroidal graphs
本文研究环面图在缺陷值不全相等条件下的不当染色问题，证明了环面图满足若干新的缺陷染色结果，并将 $(0,0,0,14)$-可染改进为 $(0,0,0,4)$-可染，同时刻画了 $6$-正则环面图及可嵌入 $N_1$ 的图的可染性。
Variable degeneracy on toroidal graphs
本文研究环面图与平面图的严格$f$-退化横贯，通过结构分析证明若干禁止构型下图的DP-$4$-可染色性及列表顶点树性度至多2，改进了已有结果。
The gonality of chess graphs
该研究从芯片发射博弈与图gonality角度出发，推导王、象、马等国际象棋图及其环面版本的gonality上下界，并证明除特定四类图外，所有国际象棋图的gonality存在仅由$\n\min\{m,n\}$决定的上界。
Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs
该文研究含单位元交换环的素理想和图的图论性质与环代数性质的关联，分类了交叉帽数至多2、亏格为1的非局部交换Artin环，证明不存在素理想和图为射影平面的这类环。
The game of cops and robbers is a well-known game played on graphs. In this paper we consider the straight-ahead orientations of 4-regular quadrangulations of the torus and the Klein bottle and we prove that their cop number is bounded by a constant. We also show that the cop number of every k-regularly oriented toroidal grid is at most 13.
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