Related papers: Edge-maximal graphs on orientable and some non-ori…

A graph is IC-planar if it admits a drawing in the plane with at most one crossing per edge and such that two pairs of crossing edges share no common end vertex. IC-planarity specializes both NIC-planarity, which allows a pair of crossing…

For a class $\mathcal{D}$ of drawings of loopless (multi-)graphs in the plane, a drawing $D \in \mathcal{D}$ is \emph{saturated} when the addition of any edge to $D$ results in $D' \notin \mathcal{D}$ - this is analogous to saturated graphs…

It is shown that a simple graph which is embeddable in the real projective plane is minimally 3-rigid if and only if it is (3,6)-tight. Moreover the topologically uncontractible embedded graphs of this type are constructible from one of 8…

In 1965, Vizing proved that every planar graph $G$ with maximum degree $\Delta\geq 8$ is edge $\Delta$-colorable. It is also proved that every planar graph $G$ with maximum degree $\Delta=7$ is edge $\Delta$-colorable by Sanders and Zhao,…

The minimal infinitesimal rigidity of bar-joint frameworks in the non-Euclidean spaces (R^2, ||.||_q) are characterised in terms of (2,2)-tight graphs. Specifically, a generically placed bar-joint framework (G,p) in the plane is minimally…

A drawing of a graph in the plane is called 1-planar if each edge is crossed at most once. A graph together with a 1-planar drawing is a 1-plane graph. A 1-plane graph $G$ with exactly $4|V (G)|-8$ edges is called optimal. The crossing…

it is shown that each triangle-free 1-planar graph with maximum degree $\Delta\geq7$ can be $\Delta$-colorable by Discharging Method.

In this paper we study some geometric properties of surfaces in the Heisenberg group, $\mathcal{H}_{3}.$ We obtain, using the Gauss map for Lie groups, a partial classification of minimal graphs in $\mathcal{H}_{3}.$ We also proof the non…

Let $G$ be a graph and $\gamma (G)$ denote the domination number of $G$, i.e. the cardinality of a smallest set of vertices $S$ such that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. Matheson and Tarjan conjectured…

Partial edge drawing (PED) is a drawing style for non-planar graphs, in which edges are drawn only partially as pairs of opposing stubs on the respective end-vertices. In a PED, by erasing the central parts of edges, all edge crossings and…

Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded…

We consider the family of graphs whose vertex set is $\mathbb{Z}^n$ where two vertices are connected by an edge when their $\ell_\infty$-distance is 1. Towards an edge isoperimetric inequality for this graph, we calculate the edge boundary…

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,…

Given a graph $G$, a Berge copy of $G$ is a hypergraph obtained by enlarging the edges arbitrarily. Gy\H ori in 2006 showed that for $r=3$ or $r=4$, an $r$-uniform $n$-vertex Berge triangle-free hypergraph has at most $\lfloor…

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…

A k-regular planar graph G is nearly Platonic when all faces but one are of the same degree while the remaining face is of a different degree. We show that no such graphs with connectivity one can exist. This complements a recent result by…

Let $(S,L)$ be a polarized K3 surface with $\mathrm{Pic}(S) = \mathbb{Z}[L]$ and $L\cdot L=2g-2$, let $C$ be a nonsingular curve of genus $g-1$ and let $f:C\to S$ be such that $f(C) \in \vert L \vert$. We prove that the Gaussian map…

An edge-girth-regular graph $egr(n,k,g,\lambda)$ is a $k-$regular graph of order $n$, girth $g$ and with the property that each of its edges is contained in exactly $\lambda$ distinct $g-$cycles. We present new families of edge-girth…

We introduce the family of $k$-gap-planar graphs for $k \geq 0$, i.e., graphs that have a drawing in which each crossing is assigned to one of the two involved edges and each edge is assigned at most $k$ of its crossings. This definition is…

Let $\mathbb{S}_g$ be the orientable surface of genus $g$. We show that the number of vertex-labelled cubic multigraphs embeddable on $\mathbb{S}_g$ with $2n$ vertices is asymptotically $c_g n^{5(g-1)/2-1}\gamma^{2n}(2n)!$, where $\gamma$…