Related papers: Toroidal Cartesian Products Where One Factor is 3-…

The well-known twenty types of 2-uniform tilings of the plane give rise infinitely many doubly semi-equivelar maps on the torus. In this article, we show that every such doubly semi-equivelar map on the torus contains a Hamiltonian cycle.…

Let $G$ be a graph and $S \subseteq V(G)$. In the cycle convexity, we say that $S$ is \textit{cycle convex} if for any $u\in V(G)\setminus S$, the induced subgraph of $S\cup\{u\}$ contains no cycle that includes $u$. The \textit{cycle…

For an action of a compact torus $T$ on a smooth compact manifold~$X$ with isolated fixed points the number $\frac{1}{2}\dim X-\dim T$ is called the complexity of the action. In this paper we study certain examples of torus actions of…

A torus-covering $T^2$-link of degree $n$ is a surface-link consisting of tori, in the form of an unbranched covering of degree $n$ over the standard torus. We focus on a torus-covering $T^2$-link of degree 3, which is determined by a pair…

We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1-factor contraction. This correspondence is…

Let $G$ and $H$ be two groups acting on path connected topological spaces $X$ and $Y$ respectively. Assume that $H$ is finite of order $m$ and the quotient maps $p:X\to X/G$ and $q:Y\to Y/H$ are regular coverings. Then it is well-known that…

Let $K\subseteq S^3$ be a knot with exterior $E_K$, and denote by $\rho\colon \pi_1(E_K)\twoheadrightarrow G$ a quotient of its group. We give a sharp obstruction to the existence of a connected, oriented, smooth surface $F\subseteq B^4$…

Let $P$ be a cubic $3$-connected bipartite plane graph which has a $2$-factor which consists only of facial $4$-cycles, and suppose that $P^{*}$ is the dual graph. We show that $P$ has at least $3^{\frac{2|P^{*}|}{\Delta^{2}{(P^{*})}}}$…

Cai et al.\ proved that a toroidal graph $G$ without $6$-cycles is $5$-choosable, and proposed the conjecture that $\textsf{ch}(G) = 5$ if and only if $G$ contains a $K_{5}$ [J. Graph Theory 65 (2010) 1--15], where $\textsf{ch}(G)$ is the…

It is well-known that Chv\'{a}tal and Erd\H{o}s stated that any graph of order at least three whose independence number is no greater than its connectivity is Hamiltonian; that any graph whose independence number is no greater than its…

Let $G=(V,E)$ be a bipartite graph embedded in a plane (or $n$-holed torus). Two subgraphs of $G$ differ by a {\it $Z$-transformation} if their symmetric difference consists of the boundary edges of a single face---and if each subgraph…

The Cartesian product of two cycles (of length m and length n) has a natural embedding on the torus, such that each face of the embedding is a 4-cycle. The toroidal grid Qd(m,n,r) is a generalization of this in which there is a shift by r…

An outer-connected dominating set for an arbitrary graph $G$ is a set $\tilde{D} \subseteq V$ such that $\tilde{D}$ is a dominating set and the induced subgraph $G [V \setminus \tilde{D}]$ be connected. In this paper, we focus on the…

Let $G$ be an undirected simple connected graph. We say a vertex $u$ is eccentric to a vertex $v$ in $G$ if $d(u,v)=\max\{d(v,w): w\in V(G)\}$. The eccentric graph, $E(G)$ of $G$ is a graph defined on the same vertex set as of $G$ and two…

Luo, Tian and Wu conjectured in 2022 that for any tree $T$ with bipartition $X$ and $Y$, every $k$-connected bipartite graph $G$ with $\delta(G) \geq k + t$, where $t = \max\{|X|,|Y |\}$, contains a subtree $T' \cong T$ such that $G-V(T')$…

Let $k$ be a field. Let $A$ and $B$ be connected $N$-graded $k$-algebras. Let $C$ denote a twisted tensor product of $A$ and $B$ in the category of connected $N$-graded $k$-algebras. The purpose of this paper is to understand when $C$…

To an oriented link in a solid torus we associate a trace graph in a thickened torus in such a way that links are isotopic if and only if their trace graphs can be related by moves of finitely many standard types. The key ingredient is a…

An integral convex polytope ${\mathcal P}$ is said to be Gorenstein if its toric ring $K[{\mathcal P}]$ is normal and Gorenstein. In this paper, Gorenstein cut polytopes of graphs are characterized explicitly. First, we prove that…

Let X=GM be a finite group factorisation. It is shown that the quantum double D(H) of the associated bicrossproduct Hopf algebra $H=kM\cobicross k(G)$ is itself a bicrossproduct $kX\cobicross k(Y)$ associated to a group YX, where $Y=G\times…

Let $\overline G$ be the wonderful compactification of a simple affine algebraic group $G$ defined over $\mathbb C$ such that its center is trivial and $G\not= {\rm PSL}(2,\mathbb{C})$. Take a maximal torus $T \subset G$, and denote by…