Related papers: The flip-graph of the 4-dimensional cube is connec…

In 2007, Ando and Egawa proved a theorem which provides a lower bound on the number of contractible edges preserving $4$-connectedness in $4$-connected graphs. In this paper, we refine their bounds, especially for the $4$-connected plane…

A graph is said to be symmetric if its automorphism group is transitive on its arcs. Guo et al. (Electronic J. Combin. 18, \#P233, 2011) and Pan et al. (Electronic J. Combin. 20, \#P36, 2013) determined all pentavalent symmetric graphs of…

It is shown that exactly 7 distance-transitive cubic graphs among the existing 12 possess a particular ultrahomogeneous property with respect to oriented cycles realizing the girth that allows the construction of a related Cayley digraph…

We prove that a connected bipartite graph G is a partial cube if and only if the set of attaching points of any copoint of G is convex. A consequence of this result is that any connected bipartite graph with pre-hull number at most 1 is a…

A zigzag in a map (a $2$-cell embedding of a connected graph in a connected closed $2$-dimensional surface) is a cyclic sequence of edges satisfying the following conditions: 1) any two consecutive edges lie on the same face and have a…

We introduce the polygonalisation complex of a surface, a cube complex whose vertices correspond to polygonalisations. This is a geometric model for the mapping class group and it is motivated by works of Harer, Mosher and Penner. Using…

For an integer $n\geq 2$, the triangular graph has vertex set the $2$-subsets of $\{1,\ldots,n\}$ and edge set the pairs of $2$-subsets intersecting at one point. Such graphs are known to be halved graphs of bipartite rectagraphs, which are…

This study is the $4^{th}$ part of a detailed study on Type-2 isomorphic circulant graphs having ten parts \cite{v2-1}-\cite{v2-10} and is a continuation of Part 3. Here, we study Type-2 isomorphic circulant graphs of order 54 and show that…

Any two triangulations of a closed surface with the same number of vertices can be transformed into each other by a sequence of regular flips, provided the number of vertices exceeds a number N depending on the surface. Examples show that…

For a planar graph with a given f-vector $(f_{0}, f_{1}, f_{2}),$ we introduce a cubic polynomial whose coefficients depend on the f-vector. The planar graph is said to be real if all the roots of the corresponding polynomial are real. Thus…

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called \emph{pseudo-simultaneously flippable edges}. Such edges are related to the notion of convex…

We characterise the structure of those graphs of a given order which maximise the number of connected induced subgraphs for seven different graph classes, each with other prescribed parameters like minimum degree, independence number,…

We observe that over an algebraically closed field, any finite-dimensional algebra is the endomorphism algebra of an m-cluster-tilting object in a triangulated m-Calabi-Yau category, where m is any integer greater than 2.

Using existing classification results for the 7- and 8-cycles in the pancake graph, we determine the number of permutations that require 4 pancake flips (prefix reversals) to be sorted. A similar characterization of the 8-cycles in the…

We provide a unique decomposition of every 4-connected graph into parts that are either quasi-5-connected, cycles of triangle-torsos and 3-connected torsos on $\leq 5$ vertices, generalised double-wheels, or thickened $K_{4,m}$'s. The…

A cube is an 8-rep-tile: it is the union of eight smaller copies of itself. Is there a set with a hole which has this property? The computer found an interesting and complicated solution, which then could be simplified. We discuss some…

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This second one applies the powerful tool of trigonometric Diophantine equations to classify the case of…

A {\em $1-$vertex triangulation} of an oriented compact surface $S$ of genus $g$ is an embedded graph $T\subset S$ with a unique vertex such that all connected components of $S\setminus T$ are triangles (adjacent to exactly 3 edges of $T$).…

There are totally 11 kinds of Archimedean tiling for the plane. Applying the Floquet-Bloch theory, we derive the dispersion relations of the periodic quantum graphs associated with a number of Archimedean tiling, namely the triangular…

The $n$-cube graph is the graph on the vertex set of $n$-tuples of $0$s and $1$s, with two vertices joined by an edge if and only if the $n$-tuples differ in exactly one component. We compute the Smith group of this graph, or, equivalently,…