Related papers: On smallest $3$-polytopes of given graph radius

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph…

The Gram dimension $\gd(G)$ of a graph is the smallest integer $k \ge 1$ such that, for every assignment of unit vectors to the nodes of the graph, there exists another assignment of unit vectors lying in $\oR^k$, having the same inner…

Recent literature posed the problem of characterising the graph degree sequences with exactly one $3$-polytopal (i.e. planar, $3$-connected) realisation. This seems to be a difficult problem in full generality. In this paper, we…

For an integer $r>0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to vertices with at least $min\{r, d(v)\}$ different colors.…

Sturmfels-Sullivant conjectured that the cut polytope of a graph is normal if and only if the graph has no K_5 minor. In the present paper, it is proved that the normality of cut polytopes of graphs is a minor closed property. By using this…

Assume that $K_{j\times n}$ be a complete, multipartite graph consisting of $j$ partite sets and $n$ vertices in each partite set. For given graphs $G_1$ and $G_2$, the multipartite Ramsey number (M-R-number) $m_j(G_1, G_2)$ is the smallest…

A connected graph is 2K2-free if it does not contain a pair of independent edges as an induced subgraph. In this paper, we present the structural characterization of minimal vertex separator and show that there are polynomial number of…

The boxicity of a graph is the smallest dimension $d$ allowing a representation of it as the intersection graph of a set of $d$-dimensional axis-parallel boxes. We present a simple general approach to determining the boxicity of a graph…

A lattice $(d,k)$-polytope is the convex hull of a set of points in $\mathbb{R}^d$ whose coordinates are integers ranging between $0$ and $k$. We consider the smallest possible distance $\varepsilon(d,k)$ between two disjoint lattice…

The $2$-packing number $\rho_2(G)$ of a graph $G$ is the cardinality of a largest $2$-packing of $G$ and the open packing number $\rho^{\rm o}(G)$ is the cardinality of a largest open packing of $G$, where an open packing (resp.…

A connected graph $G$ with at least two vertices is matching covered if each of its edges lies in a perfect matching. A matching covered graph is minimal if the removal of any edge results in a graph that is no longer matching covered. An…

Symmetric edge polytopes of graphs are important object in Ehrhart theory,and have an application to Kuramoto models. In the present paper, we study the upper and lower bounds for the number of facets of symmetric edge polytopes of…

Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers $d_0$ and $d_1$ such that planar graphs without $\{4,5\}$-cycles and $d^{\Delta}\ge d_0$ are $3$-choosable and planar graphs without $\{4,5,6\}$-cycles and…

In 2010, Koolen and Bang proposed the following conjecture: For a fixed integer $m \geq 2$, any geometric distance-regular graph with smallest eigenvalue $-m$, diameter $D \geq 3$ and $c_2 \geq 2$ is either a Johnson graph, a Grassmann…

The theorem of Chung, Graham, and Wilson on quasi-random graphs asserts that of all graphs with edge density p, the random graph G(n,p) contains the smallest density of copies of K_{t,t}, the complete bipartite graph of size 2t. Since…

An abstract polytope is \emph{flat} if every facet is incident on every vertex. In this paper, we prove that no chiral polytope has flat finite regular facets and finite regular vertex-figures. We then determine the three smallest non-flat…

Let $\mathcal{H}$ be a 3-uniform hypergraph. The multicolor Ramsey number $ r_k(\mathcal{H})$ is the smallest integer $n$ such that every coloring of $ \binom{[n]}{3}$ with $k$ colors has a monochromatic copy of $\mathcal{H}$. Let $…

It will be proved that a $k$-clique in the $1$-skeleton of either the order polytope or the chain polytope corresponds to the $(k-1)$-face, which is a simplex, in each polytope. These results generalize the known explicit descriptions of…

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple graph with vertex set $G$, in which two distinct vertices are adjacent if one of them is a power of the other. For an integer $n\geq 2$, let $C_n$ denote the cyclic group…

This paper deals with the three types of regular polytopes which exist in all dimensions -- regular simplices, cubes and regular cross-polytopes -- and their outer and inner radii. While the inner radii of regular simplices are well…