Silviu Radu
In this paper we give a upper bound of 40 on Rubik's cube group. The previously known upper bound has been 42 since 1995. In order to prove our claim we use computational methods. The program used is GAP computer algebra. Further more we give the number of elements of given length in two important subgroups of Rubik's cube group.
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We describe in details the nxnxn Rubik's Cube, namely a Rubik's Cube with n rotating slices in each face. Then we state and prove the "first law of Cubology", i.e. the solvability criterion, for it
The Rubik's cube was invented in 1974 by Erno Rubik, who had no idea of the incredible popularity and mathematical fascinations his toy would bring. Through the years of study on the mathematical properties of the cube, the Rubik's Cube group was introduced to represent all possible moves one could perform on the cube. In this paper, we define a planar analogue to the Rubik's cube, which we dub the Rubik's Square, and prove that the Rubik's square is complete in the sense that given any two configurations there is a sequence of moves which changes one to the other. The Rubik's cube does not ha…
The Rubik's Cube is the most popular puzzle in the world. Two of its studied aspects are God's Number, the minimum number of turns necessary to solve any state, and the first law of cubology, a solvability criterion. We modify previous statements of the first law of cubology for $n \times n \times n$ Rubik's Cubes, and prove necessary and sufficient solvability conditions. We compute the order of the Rubik's Cube group and the number of distinct configurations of the $n \times n \times n$ Rubik's Cube. Finally, we derive a lower bound for God's Number using the group theoretical results and a …
The Rubik's cube is a famous puzzle in which faces can be moved and the corresponding movement operations define a group. We consider here a generalization to any $3$-valent map. We prove an upper bound on the size of the corresponding group which we conjecture to be tight.
In this paper, we present a novel algorithm and its three variations for solving the Rubik's cube more efficiently. This algorithm can be used to solve the complete $n \times n \times n$ cube in $O(\frac{n^2}{\log n})$ moves. This algorithm can also be useful in certain cases for speedcubers. We will prove that our algorithm always works and then perform a basic analysis on the algorithm to determine its algorithmic complexity of $O(n^2)$. Finally, we further optimize this complexity to $O(\frac{n^2}{\log n})$.
In this paper, we prove that optimally solving an $n \times n \times n$ Rubik's Cube is NP-complete by reducing from the Hamiltonian Cycle problem in square grid graphs. This improves the previous result that optimally solving an $n \times n \times n$ Rubik's Cube with missing stickers is NP-complete. We prove this result first for the simpler case of the Rubik's Square---an $n \times n \times 1$ generalization of the Rubik's Cube---and then proceed with a similar but more complicated proof for the Rubik's Cube case.
The Rubik's Cube is perhaps the world's most famous and iconic puzzle, well-known to have a rich underlying mathematical structure (group theory). In this paper, we show that the Rubik's Cube also has a rich underlying algorithmic structure. Specifically, we show that the n x n x n Rubik's Cube, as well as the n x n x 1 variant, has a "God's Number" (diameter of the configuration space) of Theta(n^2/log n). The upper bound comes from effectively parallelizing standard Theta(n^2) solution algorithms, while the lower bound follows from a counting argument. The upper bound gives an asymptotically…
Graph-theoretical estimates of the diameters of the Rubik's Cube groups
本文提出一种基于图阶数、度数、围长及过顶点圈数等局部参数的对称图直径严格下界方法，并应用于魔方群凯莱图，获得比随机正则图下界更紧、达到正确直径60%至77%的估计。
Rubik's Abstract Polytopes
本文将魔方及其构型群推广至任意抽象正多面体，完整刻画任意维魔方单纯形的相关群，并概述魔方超立方体的类似结论。
Solving infinitary Rubik's cubes
本文研究无穷阶魔方的可解性，区分有边与无边两种变体，证明有边魔方在少于 $ω_{α+1}$ 次操作内可解，可数无边魔方在 $ω^2$ 次操作内可解，并提出不可数无边魔方的开放问题。
A Demigod's Number for the Rubik's Cube
本文提出一种易于复现的方法，通过采样随机状态并利用点传递图平均距离与直径的关系，以高置信度证明魔方直径至多36步，并推广到一般点传递图。
Probabilistic estimates of the diameters of the Rubik's Cube groups
本文利用概率论中修改版优惠券收集者问题估计魔方群直径，验证了2×2×2和3×3×3魔方在四分之一转度量下的已知直径，并预测4×4×4和5×5×5魔方的直径分别为48（41）和68（58），同时给出直径的近似公式。
Let $Γ$ denote a distance-regular graph. The maximum size of codewords with minimum distance at least $d$ is denoted by $A(Γ,d)$. Let $\square_n$ denote the folded $n$-cube $H(n,2)$. We give an upper bound on $A(\square_n,d)$ based on block-diagonalizing the Terwilliger algebra of $\square_n$ and on semidefinite programming.The technique of this paper is an extension of the approach taken by A. Schrijver \cite{s} on the study of $A(H(n,2),d)$.
A {\it good drawing} of a graph $G$ is a drawing where the edges are non-self-intersecting and each two edges have at most one point in common, which is either a common end vertex or a crossing. The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a good drawing of $G$ in the plane. The {\it $n$-dimensional augmented cube} $AQ_n$, proposed by S.A. Choudum and V. Sunitha, is an important interconnection network with good topological properties and applications. In this paper, we obtain an upper bound on the crossing number of $AQ_n$ less than $26/…
The {\it crossing number} of a graph $G$ is the minimum number of pairwise intersections of edges in a drawing of $G$. Motivated by the recent work [Faria, L., Figueiredo, C.M.H. de, Sykora, O., Vrt'o, I.: An improved upper bound on the crossing number of the hypercube. J. Graph Theory {\bf 59}, 145--161 (2008)] which solves the upper bound conjecture on the crossing number of $n$-dimensional hypercube proposed by Erdős and Guy, we give upper and lower bounds of the crossing number of locally twisted cube, which is one of variants of hypercube.
The balanced upper chromatic number of linear hypergraphs and the $n$-cube over $t$ elements
本文研究线性超图与n-立方体$C_t^n$的平衡上色数，针对任意线性超图给出该参数的通用上界，确定$t\geq4n-2$时$C_t^n$的精确值，并在2、3维情形下给出更小t值时的相关结果。
We obtain a series of lower bounds for the product set of combinatorial cubes, as well as some non--trivial upper estimates for the multiplicative energy of such sets.
The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of order N contains the cube graph Q_n or an independent set of order s. Burr and Erdos in 1983 asked whether the simple lower bound r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.
Cop numbers for subclasses of partial cubes
该研究针对部分立方体子类，改进了中位数图的警察数上界，建立了单纯复形图警察数的上下界，并确定了二分轮图等特殊单纯复形图的警察数精确值。
For every even positive integer $k\ge 4$ let $f(n,k)$ denote the minimim number of colors required to color the edges of the $n$-dimensional cube $Q_n$, so that the edges of every copy of $k$-cycle $C_k$ receive $k$ distinct colors. Faudree, Gyárfás, Lesniak and Schelp proved that $f(n,4)=n$ for $n=4$ or $n>5$. We consider larger $k$ and prove that if $k \equiv 0$ (mod 4), then there are positive constants $c_1, c_2$ depending only on $k$ such that $$c_1n^{k/4} < f(n,k) < c_2 n^{k/4}.$$ Our upper bound uses an old construction of Bose and Chowla of generalized Sidon sets. For $k \equiv 2$ (mod…
We show upper bounds on the maximal dimension $d$ of Hilbert cubes $H=a_0+\{0,a_1\}+\cdots + \{0, a_d\}\subset S \cap [1, N]$ in several sets $S$ of arithmetic interest such as the squares, powerful numbers and pure powers.
Burning numbers via eigenpolytopes -- Hamming graphs, Johnson graphs, and halved cubes
该研究针对汉明图、约翰逊图和半立方体图的燃烧数问题，利用特征多面体1-骨架与原图同构的性质给出上下界，并基于Alon的超立方体燃烧数研究思想，提出特征多面体动态搜索算法寻找未燃烧顶点。
Let $Q_n$ be the $n$-dimensional Hamming cube and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically \[2n2^{N/4},\] as was conjectured by Ilinca and the first author in connection with a question of Duffus, Frankl and Rödl. The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof that it is also an upper bound draws on various tools, among them "stability" results for maximal independent set counts and old and new results on isoperimetric behavior in $Q_n$.
Cop number of partial cubes
本文研究部分立方体（超立方体的等距子图）的警察数，建立其下界，对子类给出上界，并针对斐波那契立方体改进这些界，该图类与化学图论中的共振图模型相关。
We obtain upper bounds on the cardinality of Hilbert cubes in finite fields, which avoid large product sets and reciprocals of sum sets. In particular, our results replace recent estimates of N. Hegyvári and P. P. Pach (2020), which appear to be void for all admissible parameters. Our approach is different from that of N. Hegyvári and P. P. Pach and is based on some well-known bounds of double character and exponential sums over arbitrary sets, due to A. A. Karatsuba (1991) and N. G. Moshchevitin (2007), respectively.
Let $Q_n$ be the poset that consists of all subsets of a fixed $n$-element set, ordered by set inclusion. The poset cube Ramsey number $R(Q_n,Q_n)$ is defined as the least $m$ such that any 2-coloring of the elements of $Q_m$ admits a monochromatic copy of $Q_n$. The trivial lower bound $R(Q_n,Q_n)\ge 2n$ was improved by Cox and Stolee, who showed $R(Q_n,Q_n)\ge 2n+1$ for $3\le n\le 8$ and $n\ge 13$ using a probabilistic existence proof. In this paper, we provide an explicit construction that establishes $R(Q_n,Q_n)\ge 2n+1$ for all $n\ge 3$. The best known upper bound, due to Lu and Thompson,…
An axis-parallel b-dimensional box is a Cartesian product $R_1 \times R_2 \times ... \times R_b$ where each $R_i$ (for $1 \leq i \leq b$) is a closed interval of the form $[a_i,b_i]$ on the real line. The boxicity of any graph $G$, box(G) is the minimum positive integer b such that G can be represented as the intersection graph of axis parallel b-dimensional boxes. A b-dimensional cube is a Cartesian product $R_1 \times R_2\times ... \times R_b$, where each $R_i$ (for $1 \leq i \leq b$) is a closed interval of the form [$a_i$,$a_i$+1] on the real line. When the boxes are restricted to be axis-…
We study the dispersion of a point set, a notion closely related to the discrepancy. Given a real $r\in (0,1)$ and an integer $d\geq 2$, let $N(r,d)$ denote the minimum number of points inside the $d$-dimensional unit cube $[0,1]^d$ such that they intersect every axis-aligned box inside $[0,1]^d$ of volume greater than $r$. We prove an upper bound on $N(r,d)$, matching a lower bound of Aistleitner et al. up to a multiplicative constant depending only on $r$. This fully determines the rate of growth of $N(r,d)$ if $r\in(0,1)$ is fixed.
An upper bound on the number of frequency hypercubes
本文通过构造规模为 $s^n$（$s<q-1$）的更小测试集，改进了频率 $n$-立方体数量的上界，并为广义频率 $n$-立方体（相关免疫函数）构造了基数更小的新测试集。
In the d-dimensional online bin packing problem, d-dimensional cubes of positive sizes no larger than 1 are presented one by one to be assigned to positions in d-dimensional unit cube bins. In this work, we provide improved upper bounds on the asymptotic competitive ratio for square and cube bin packing problems, where our bounds do not exceed 2.0885 and 2.5735 for square and cube packing, respectively. To achieve these results, we adapt and improve a previously designed harmonic-type algorithm, and apply a different method for defining weight functions. We detect deficiencies in the state-of-…
The pebbling number of Fibonacci cubes
本文确定了Fibonacci立方体Γ_n的推石数π(Γ_n)等于2^n，尽管Γ_n的顶点数远少于超立方体Q_n。研究通过势函数论证得出下界，并利用加权函数引理和穷举MILP验证得出上界，从而缩小了差距。
A note on cube-free problems
本文研究了无立方体子集的大小上界问题，证明了在Z_N中2N/3的上界成立且当N被3整除时紧致，提出了d-无立方体子集的猜想并在最小素因子或素数幂等情形下证明，同时探讨了相关的无集合问题。
A $d$-dimensional polycube is a facet-connected set of cells (cubes) on the $d$-dimensional cubical lattice $\mathbb{Z}^d$. Let $A_d(n)$ denote the number of $d$-dimensional polycubes (distinct up to translations) with $n$ cubes, and $λ_d$ denote the limit of the ratio $A_d(n{+}1)/A_d(n)$ as $n \to \infty$. The exact value of $λ_d$ is still unknown rigorously for any dimension $d \geq 2$; the asymptotics of $λ_d$, as $d \to \infty$, also remained elusive as of today. In this paper, we revisit and extend the approach presented by Klarner and Rivest in 1973 to bound $A_2(n)$ from above. Our cont…
Sharp Bounds for Discrete Cube Skeleta
研究确定包含特定立方体骨架的有限格集最小阶数问题，核心方法是结合中点估计、投影不等式及强归纳法，得出对于\(N\)个中心答案为\(N^{1-(n - k)/(2n^2)}\)，如含正方形边界的格集至少有常数倍\(N^{7/8}\)个点。
A quadratic upper bound on the Chvátal rank of polytopes in the 0/1-cube
本文证明0/1立方体中多面体的Chvátal秩至多为二次阶，改进先前上界，并借助多尺度Dirichlet逼近定理确立其紧的$\Theta(n^2)$复杂度。
A Generalisation of the Concentration-of-Measure Phenomenon with Applications to Intersection Problems
本文在离散立方体上推广了浓度测度现象，定义了“(γ,C)-良好散布”子集，并证明对于任何大子集A和良好散布集S，A+S的测度接近全空间。利用该结果，对C4-交集问题和4-AP-交集问题给出了非平凡上界，并对H-交集问题和k-AP-交集问题给出了趋于0的上界。
Let $T_ε$ be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a nonnegative function on $\{0,1\}^n$ and let $q \ge 1$. We upper bound the $\ell_q$ norm of $T_ε f$ by the average $\ell_q$ norm of conditional expectations of $f$, given sets of roughly $(1-2ε)^{r(q)} \cdot n$ variables, where $r$ is an explicitly defined function of $q$. We describe some applications for error-correcting codes and for matroids. In particular, we derive an upper bound on the weight distribution of duals of BEC-capacity achieving binary linear codes. This improves the known bounds …
We give a new upper bound for the average graph distance in terms of the average Ollivier curvature. Here, the average Ollivier curvature is weighted with the edge betweenness centrality. Moreover, we prove that equality is attained precisely for the reflective graphs which have been classified as Cartesian products of cocktail party graphs, Johnson graphs, halved cubes, Schläfli graphs, and Gosset graphs.
We prove that the Random-Edge simplex algorithm requires an expected number of at most 13n/sqrt(d) pivot steps on any simple d-polytope with n vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial d-cubes, the trivial upper bound of 2^d on the performance of Random-Edge can asymptotically be improved by any desired polynomial factor in d.
Counting and Covering in Nearest-Neighbour Representations of Boolean Functions
本研究探讨布尔函数最近邻表示所需原型数，强化下界并给出VC维界，通过覆盖设计精确刻画对称阈值函数，并证明随机函数需约半数原型，对称函数需11/20。
Let $A(n,d)$ (respectively $A(n,d,w)$) be the maximum possible number of codewords in a binary code (respectively binary constant-weight $w$ code) of length $n$ and minimum Hamming distance at least $d$. By adding new linear constraints to Schrijver's semidefinite programming bound, which is obtained from block-diagonalising the Terwilliger algebra of the Hamming cube, we obtain two new upper bounds on $A(n,d)$, namely $A(18,8) \leq 71$ and $A(19,8) \leq 131$. Twenty three new upper bounds on $A(n,d,w)$ for $n \leq 28$ are also obtained by a similar way.
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