Related papers: A full Halin grid theorem

Z-mapping graph is a balanced bipartite graph $G$ of a digraph $D$ by split each vertex of $D$ into a pair of vertices of $G$. Based on the property of the $G$, it is proved that if $D$ is strong connected and $G$ is Hamiltonian, then $D$…

A graph $G$ is perfectly divisible if, for every induced subgraph $H$ of $G$, either $V(H)$ is a stable set or admits a partition into two sets $X_1$ and $X_2$ such that $\omega(H[X_1]) < \omega(H)$ and $H[X_2]$ is a perfect graph. In this…

Let $G$ be a graph of order $n$. A classical upper bound for the domination number of a graph $G$ having no isolated vertices is $\lfloor\frac{n}{2}\rfloor$. However, for several families of graphs, we have $\gamma(G) \le…

In this series of papers, the primary goal is to enumerate Hamiltonian cycles (HC's) on the grid cylinder graphs $P_{m+1}\times C_n$, where $n$ is allowed to grow whilst $m$ is fixed. In Part~I, we studied the so-called non-contractible…

We say that a graph G is $(k,\ell)$-stable if removing $k$ vertices from it reduces its independence number by at most $\ell$. We say that G is tight $(k,\ell)$-stable if it is $(k,\ell)$-stable and its independence number equals…

A good edge-labelling of a simple graph is a labelling of its edges with real numbers such that, for any ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. Say a graph is good if it admits a good…

This article investigates the properties of order-divisor graphs associated with finite groups. An order-divisor graph of a finite group is an undirected graph in which the set of vertices includes all elements of the group, and two…

We study matching-removability under the degree/connectivity regime of Halin's theorem, which asserts that every $k$-connected graph $G$ with minimum degree $\delta(G)\ge k+1$ contains an edge $e$ such that $G-e$ remains $k$-connected. For…

Let $\mathbf G$ be a graphing, that is a Borel graph defined by $d$ measure preserving involutions. We prove that if $\mathbf G$ is {\em treeable} then it arises as the local limit of some sequence $(G_n)_{n\in\mathbb{N}}$ of graphs with…

A circular-arc hypergraph $H$ is a hypergraph admitting an arc ordering, that is, a circular ordering of the vertex set $V(H)$ such that every hyperedge is an arc of consecutive vertices. An arc ordering is tight if, for any two hyperedges…

Let $k$ be an integer. We prove a rough structure theorem for separations of order at most $k$ in finite and infinite vertex transitive graphs. Let $G = (V,E)$ be a vertex transitive graph, let $A \subseteq V$ be a finite vertex-set with…

Given $\alpha>0$ and an integer $\ell\geq5$, we prove that every sufficiently large $3$-uniform hypergraph $H$ on $n$ vertices in which every two vertices are contained in at least $\alpha n$ edges contains a copy of $C_\ell^{-}$, a tight…

A graph is said to be globally rigid in $d$-dimensional space if almost all of its embeddings are unique up to isometries. If a graph has enough automorphisms to send any of its vertices into any other, then it is called vertex-transitive.…

One way to certify that a graph does not contain an induced cycle of length six is to provide a partition of its vertex set into (i) a stable set, and (ii) a graph containing no stable set of size three and no induced matching of size two.…

A bond in a graph is a minimal nonempty edge-cut. A connected graph $G$ is dual Hamiltonian if the vertex set can be partitioned into two subsets $X$ and $Y$ such that the subgraphs induced by $X$ and $Y$ are both trees. There is much…

Let $H$ be a fixed graph. What can be said about graphs $G$ that have no subgraph isomorphic to a subdivision of $H$? Grohe and Marx proved that such graphs $G$ satisfy a certain structure theorem that is not satisfied by graphs that…

This work proposes \textsc{H-Td}, a practical linear-time algorithm for computing an optimal-width tree decomposition of Halin graphs. Unlike state-of-the-art methods based on reduction rules or separators, \textsc{H-Td} exploits the…

In 1952, Dirac proved the following theorem about long cycles in graphs with large minimum vertex degrees: Every $n$-vertex $2$-connected graph $G$ with minimum vertex degree $\delta\geq 2$ contains a cycle with at least $\min\{2\delta,n\}$…

We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.

Let $H$ be a graph with $\Delta(H) \leq 2$, and let $G$ be obtained from $H$ by gluing in vertex-disjoint copies of $K_4$. We prove that if $H$ contains at most one odd cycle of length exceeding $3$, or if $H$ contains at most $3$…