Richard Leyland
A Hamiltonian embedding is an embedding of a graph $G$ such that the boundary of each face is a Hamiltonian cycle of $G$. It is shown that the hypercube graph $Q_n$ admits such an embedding on an orientable surface when $n$ is a power of 2. Basic necessary conditions on Hamiltonian embeddings for $Q_n$ and conjectures are made about other values of $n$.
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The balanced hypercube $BH_{n}$, a variant of the hypercube, is a novel interconnection network for massive parallel systems. It is known that the balanced hypercube remains Hamiltonian after deleting at most $4n-5$ faulty edges if each vertex is incident with at least two edges in the resulting graph for all $n\geq2$. In this paper, we show that there exists a fault-free Hamiltonian cycle in $BH_{n}$ for $n\ge 2$ with $\left | F \right |\le 5n-7$ if the degree of every vertex in $BH_{n}-F$ is at least two and there exists no $f_{4}$-cycles in $BH_{n}-F$, which improves some known results.
The balanced hypercube, $BH_n$, is a variant of hypercube $Q_n$. Zhou et al. [Inform. Sci. 300 (2015) 20-27] proposed an interesting problem that whether there is a fault-free Hamiltonian cycle in $BH_n$ with each vertex incident to at least two fault-free edges. In this paper, we consider this problem and show that each fault-free edge lies on a fault-free Hamiltonian cycle in $BH_n$ after no more than $4n-5$ faulty edges occur if each vertex is incident with at least two fault-free edges for all $n\ge 2$. Our result is optimal with respect to the maximum number of tolerated edge faults.
Alphatrion conjectured that it is possible to label the vertices of an $n$-dimensional hypercube with distinct positive integers such that for every Hamiltonian path $a_1, \dots, a_{2^n},$ we have $a_i + a_{i+1}$ prime for all $i.$ We prove the conjecture by proving the more general result that a graph $G = (V, E)$ can be labeled with distinct positive integers such that the edge sum for all $e \in E$ is prime if and only if $G$ is bipartite.
Hamiltonian laceability with a set of faulty edges in hypercubes
本文研究了超立方体在特定条件故障边下的哈密顿性质，证明了当故障边数不超过 \\( 4n-17 \\) 且满足特定度数条件时，不同部集顶点间存在哈密顿路径，揭示了超立方体网络的容错特性。
Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension edge $\edg{v_1}{v_2}$ is the number of 1's modulus 2 of any of its vertex ignoring the $i$-th entry. We prove that the number of $i$-th dimension edges appearing in a given Hamiltonian cycle of $Q_n$ with parity zero coincides with the number of edges with parity one. As an application of this result it is introduced and explored the conjecture of the inscribed…
Hamiltonian paths extending a set of matchings in hypercubes
本文研究了超立方体中扩展匹配集的哈密顿路径问题，证明了对于 n ≥ 5 且匹配集大小不超过 3n - 13 时，存在连接相对二部集顶点且包含该匹配集的哈密顿路径。
Aims: Try to prove the $n$-dimensional balanced hypercube $BH_n$ is $(2n-2)$-fault-tolerant-prescribed hamiltonian laceability. Methods: Prove it by induction on $n$. It is known that the assertation holds for $n\in\{1,2\}$. Assume it holds for $n-1$ and prove it holds for $n$, where $n\geq 3$. If there are $2n-3$ faulty links and they are all incident with a common node, then we choose some dimension such that there is one or two faulty links and no prescribed link in this dimension; Otherwise, we choose some dimension such that the total number of faulty links and prescribed links does not e…
Ruskey and Savage asked the following question: Does every matching of $Q_{n}$ for $n\geq2$ extend to a Hamiltonian cycle of $Q_{n}$? J. Fink showed that the question is true for every perfect matching, and solved the Kreweras' conjecture. In this paper we consider the question in hypercubes with faulty edges. We show that every matching $M$ of at most $2n-1$ edges can be extended to a Hamiltonian cycle of $Q_{n}$ for $n\geq2$. Moreover, we can prove that when $n\geq4$ and $M$ is nonempty this result still holds even if $Q_{n}$ has at most $n-1-\lceil\frac{|M|}{2}\rceil$ faulty edges with one …
We consider hypercubes with pairwise disjoint faulty edges. An $n$-dimensional hypercube $Q_n$ is an undirected graph with $2^n$ nodes, each labeled with a distinct binary strings of length $n$. The parity of the vertex is 0 if the number of ones in its labels is even, and is 1 if the number of ones is odd. Two vertices $a$ and $b$ are connected by the edge iff $a$ and $b$ differ in one position. If $a$ and $b$ differ in position $i$, then we say that the edge $(a,b)$ goes in direction $i$ and we define the parity of the edge as the parity of the end with 0 on the position $i$. It was already …
Szepietowski [A. Szepietowski, Hamiltonian cycles in hypercubes with $2n-4$ faulty edges, Information Sciences, 215 (2012) 75--82] observed that the hypercube $Q_n$ is not Hamiltonian if it contains a trap disconnected halfway. A proper subgraph $T$ is disconnected halfway if at least half of its nodes have parity 0 (or 1, resp.) and the edges joining all nodes of parity 0 (or 1, resp.) in $T$ with nodes outside $T$, are faulty. The simplest examples of such traps are: (1) a vertex with $n-1$ incident faulty edges, or (2) a cycle $(u,v,w,x)$, where all edges going out of the cycle from $u$ and…
On the existence of Hamiltonian cycles in hypercubes
本文研究了在k维超立方体顶点集C(2,k)中，当连续顶点间欧几里得距离固定时哈密顿回路的存在性，证明了当h为奇数且1≤h≤k-1时存在此类回路。
A perfect matching reciprocity method for embedding multiple hypercubes in an augmented cube: Applications to Hamiltonian decomposition and fault-tolerant Hamiltonicity
本文利用增强立方体的凯莱性质和互反完美匹配方法，给出AQ_n中Q_n同构子图数量的下界及最少公共边子图对的构造，并应用于证明AQ_n的边不相交哈密顿圈分解和容错双泛圈性。
Let $F_{v}$ (resp. $F_e$) be the set of faulty vertices (resp. faulty edges) in the $n$-dimensional balanced hypercube $BH_n$. Fault-tolerant Hamiltonian laceability in $BH_n$ with at most $2n-2$ faulty edges is obtained in [Inform. Sci. 300 (2015) 20--27]. The existence of edge-Hamiltonian cycles in $BH_n-F_e$ for $|F_e|\leq 2n-2$ are gotten in [Appl. Math. Comput. 244 (2014) 447--456]. Up to now, almost all results about fault-tolerance in $BH_n$ with only faulty vertices or only faulty edges. In this paper, we consider fault-tolerant cycle embedding of $BH_n$ with both faulty vertices and f…
A connected simple graph is said dual-hamiltonian if its vertex set has a $2$-coloring such that each color class induces a tree. We call such a coloring a hamiltonian coloring. We prove that if $G$ is a graph with a certain type of hamiltonian coloring and $T$ is a tree, then $G\times T$ is also dual-hamiltonian having the same certain type of hamiltonian coloring. This result is used to constructed a class of dual-hamiltonian graphs, which includes the hypercubes and other multidimensional grids.
The spined cube $SQ_n$ is a variant of the hypercube $Q_n$, introduced by Zhou et al. in [Information Processing Letters 111 (2011) 561-567] as an interconnection network for parallel computing. A graph $\G$ is an $m$-Cayley graph if its automorphism group $\Aut(\G)$ has a semiregular subgroup acting on the vertex set with $m$ orbits, and is a Caley graph if it is a 1-Cayley graph. It is well-known that $Q_n$ is a Cayley graph of an elementary abelian 2-group $\mz_2^n$ of order $2^n$. In this paper, we prove that $SQ_n$ is a 4-Cayley graph of $\mz_2^{n-2}$ when $n\geq6$, and is a $\lfloor n/2\…
A graph is said to exhibit perfect state transfer (PST) if one of its corresponding Hamiltonian matrices, which are based on the vertex-edge structure of the graph, gives rise to PST in a quantum information-theoretic context, namely with respect to inter-qubit interactions of a quantum system. We perform various perturbations to the hypercube graph---a graph that is known to exhibit PST---to create graphs that maintain many of the same properties of the hypercube, including PST as well as the distance for which PST occurs. We show that the sensitivity with respect to readout time errors remai…
Listing faces of polytopes
本文提出了一种高效算法，用于构造多面体的面哈密顿回路，解决了B-排列多面体的面哈密顿回路猜想。
A proof of Lehmer's permutation conjecture for neighbor-swap graphs
本文证明了 Lehmer 关于多重集置换的相邻交换图存在非完美哈密顿遍历的猜想，通过将单词划分为超立方体并沿生成树粘合哈密顿环，辅以两个有限显式环，并用 Python 验证和 Lean 4 形式化证明。
Hypercubes and Hamilton cycles of display sets of rooted phylogenetic networks
本文研究有根系统发育网络所嵌入树之间的组合关系，提出 rSPR 图并证明其 Hamilton 圈性质，进而给出 level-1 网络无额外推断树的充要条件及多项式重建算法。
The Pairing-Hamiltonian property in graph prisms
该研究拓展了Fink关于超立方体PH性质的结论，证明具有配对哈密顿（PH）性质的图的棱柱图也具备该性质，且连通图的足够高阶次棱柱均具有PH性质。
Hamiltonian connectivity of some base-cobase graphs
本文研究拟阵基-余基图的哈密顿连通性，利用多面体方法证明了格点路径拟阵串联并联扩展及轮与涡基-余基图的连通性，并发现正则拟阵 $R_{10}$ 提供了反例。
Note on Hamiltonicity of basis graphs of even delta-matroids
本文证明了偶 delta-matroid 基图在顶点数大于二时的 Hamiltonian 性及其环结构性质，并推广了关于 matroids 的相关经典结论。
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