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In this paper, we present the first nontrivial lower bound on the translative covering density of octahedron. To this end, we show the lower bound, in any translative covering of octahedron, on the density relative to a given parallelehedron. The resulting lower bound on the translative covering density of octahedron is $1+6.6\times10^{-8}$.

In 1980, V. I. Arnold studied the classification problem for convex lattice polygons of given area. Since then, this problem and its analogues have been studied by many authors, including $\mathrm{B\acute{a}r\acute{a}ny}$, Lagarias, Pach, Santos, Ziegler and Zong. Recently, Zong proposed two computer programs to prove Hadwiger's covering conjecture and Borsuk's partition problem, respectively, based on enumeration of the convex lattice polytopes contained in certain balls. For this purpose, similar to $\mathrm{B\acute{a}r\acute{a}ny}$ and Pach's work on volume and Liu and Zong's work on cardinality, we obtain bounds on the number of non-equivalent convex lattice polygons in a given disc. Furthermore, we propose an algorithm to enumerate these convex lattice polygons., Comment: 19 pages, 3 figures

The following assertion was equivalent to a conjecture proposed by B. Tomaszewski : Let $C$ be an $n$-dimensional unit cube and let $H$ be a plank of thickness $1$, both are centered at the origin, then no matter how to turn the cube around, $C\cap H$ contains at least half of the cube's $2^n$ vertices. A lower bound for the number of the vertices of $C$ in $C\cap H$ was obtained.

The degree sequence of a graph is the sequence of the degrees of its vertices. If $\pi$ is a degree sequence of a graph $G$, then $G$ is a realization of $\pi$ and $G$ realizes $\pi$. Determining when a sequence of positive integers is realizable as a degree sequence of a simple graph has received much attention. One of the early results, by Erd\"{o}s and Gallai, characterized degree sequences of graphs. The result was strengthened by Hakimi and Havel. Another generalization is derived by Cai et al. Hoogeveen and Sierksma listed seven criteria and gave a uniform proof. In addition, Gale and Ryser independently established a characterization by using network flows. We extend Gale and Ryser's results from bipartite graphs to tripartite graphs and even $n$-partite graphs. As corollaries, we give a necessary condition and a sufficient condition for the triple $(\sigma_1, \sigma_2, \sigma_3)$ to be realizable by a tripartite graph, where $\sigma_1$, $\sigma_2$ and $\sigma_3$ are all non-increasing sequences of nonnegative integers. We also give a stronger necessary condition for $(\sigma_1, \sigma_2, \sigma_3)$ to be realizable by a tripartite graph.

In this paper, we present the first nontrivial lower bound on the translative covering density of tetrahedra. To this end, we show the lower bound, in any translative covering of tetrahedra, on the density relative to a given cube. The resulting lower bound on the translative covering density of tetrahedra is $1+1.227\times10^{-3}$.

Let G be a vertex-colored graph. A vertex cut S of G is called a monochromatic vertex cut if the vertices of S are colored with the same color. A graph G is monochromatically vertex-disconnected if any two nonadjacent vertices of G has a monochromatic vertex cut separating them. The monochromatic vertex-disconnection number of G, denoted by mvd(G), is the maximum number of colors that are used to make G monochromatically vertex-disconnected. In this paper, the connection between the graph parameters are studied: mvd(G), connectivity and block decomposition. We determine the value of mvd(G) for some well known graphs, and then characterize G when n-5\leq mvd(G)\leq n and all blocks of G are minimally 2-connected triangle-free graphs. We obtain the maximum size of a graph G with mvd(G)=k for any k. Furthermore, we study the Erd\H{o}s-Gallai-type results for mvd(G), and completely solve them. Finally, we propose an algorithm to compute mvd(G) and give an mvd-coloring of G., Comment: arXiv admin note: text overlap with arXiv:2112.00509

Let G be a vertex-colored graph. We call a vertex cut S of G a monochromatic vertex cut if the vertices of S are colored with the same color. The graph G is monochromatically vertex-disconnected if any two nonadjacent vertices of G has a monochromatic vertex cut separating them. The monochromatic vertex-disconnection number of G, denoted by mvd(G), is the maximum number of colors that are used to make G monochromatically vertex-disconnected. In this paper, we propose an algorithm to compute mvd(G) and give an mvd-coloring of the graph G. We run this algorithm with an example written in Java. The main part of the code is shown in Appendix B and the complete code is given on Github: https://github.com/fumiaoT/mvd-coloring.git. Secondly, inspired by the previous localization principle, we obtain a upper bound of mvd(G) for some special classes of graphs. In addition, when mvd(G) is large and all blocks of G are minimally 2-connected triangle-free graphs, we characterize G. On these bases, we show that any graph whose blocks are all minimally 2-connected graphs of small order, can be computed mvd(G) and given an mvd-coloring in polynomial time.

The polygon $P$ is small if its diameter equals one. When $n=2^s$, it is still an open problem to find the maximum perimeter or the maximum width of a small $n$-gon. Motivated by Bingane's series of works, we improve the lower bounds for the maximum perimeter and the maximum width., Comment: 13 pages, 6 figures

In 1957, Hadwiger made a conjecture that every $n$-dimensional convex body can be covered by $2^n$ translations of its interior. The Hadwiger's covering functional $\gamma_m(K)$ is the smallest positive number $r$ such that $K$ can be covered by $m$ translations of $rK$. Due to Zong's program, we study the Hadwiger's covering functional for the simplex and the cross-polytope. In this paper, we give upper bounds for the Hadwiger's covering functional of the simplex and the cross-polytope., Comment: 12 pages

Let $L_1=([a_1,b_1],\ldots,[a_m,b_m])$ and $L_2=([c_1,d_1],\ldots,[c_n,d_n]$) be two sequences of intervals consisting of nonnegative integers with $b_1\ge \cdots\ge b_m$ and $d_1\ge \cdots\ge d_n$. In this paper, we first give two optimal conditions for the sequences of intervals $L_1$ and $L_2$ such that each pair $(P;Q)$ with $P=(p_1,\ldots,p_m)$, $Q=(q_1,\ldots,q_n)$, $a_i\le p_i\le b_i$ for $1\le i\le m$, $c_i\le q_i\le d_i$ for $1\le i\le n$ and $\sum\limits_{i=1}^m p_i=\sum\limits_{i=1}^n q_i$ is bigraphic. One of them is optimal sufficient condition and the other one optimal necessary condition. We also present a characterization of $(L_1;L_2)$ that is forcibly bigraphic on sequences of intervals. This is an extension of the well-known theorem on bigraphic sequences due to Gale and Ryser

In 1957, Hadwiger made the famous conjecture that any convex body of $n$-dimensional Euclidean space $\mathbb{E}^n$ can be covered by $2^n$ smaller positive homothetic copies. Up to now, this conjecture is still open for all $n\geq 3$. Denote by $\gamma_{m}(K)$ the smallest positive number $\lambda$ such that $K$ can be covered by $m$ translations of $\lambda K$. The values of $\gamma_m(K)$ for some particular $m$ and $K$ have been studied. In this article, we will focus on the situation where $K$ is the unit crosspolytope of the three-dimensional., Comment: 12 pages, 5 figures