Related papers: Constructing Two Edge-Disjoint Hamiltonian Cycles …

We consider the derangement graph in which the vertices are permutations of $\{ 1,\ldots, n\}$. Two vertices are joined by an edge if the corresponding permutations differ in every position. The derangement graph is known to be Hamiltonian…

We consider how many random edges need to be added to a graph of order $n$ with minimum degree $\alpha n$ in order that it contains the square of a Hamilton cycle w.h.p..

We provide a fast distributed algorithm for detecting $h$-cycles in the \textsf{Congested Clique} model, whose running time decreases as the number of $h$-cycles in the graph increases. In undirected graphs, constant-round algorithms are…

We give two equivalent formulations of a conjecture [2,4] on the number of arc-disjoint Hamiltonian cycles in De Bruijn graphs.

We present a Hamilton cycle in the $k$-sided pancake network and four combinatorial algorithms to traverse the cycle. The network's vertices are coloured permutations $\pi = p_1p_2\cdots p_n$, where each $p_i$ has an associated colour in…

Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices then $G$ contains at least…

We consider the problem of orienting the edges of the $n$-dimensional hypercube so only two different in-degrees $a$ and $b$ occur. We show that this can be done, for two specified in-degrees, if and only if an obvious necessary condition…

The enumeration of Hamiltonian cycles on 2n*2n grids of nodes is a longstanding problem in combinatorics. Previous work has concentrated on counting all cycles. The current work enumerates nonisomorphic cycles -- that is, the number of…

In this paper we study the connectivity properties of the random subgraph of the $n$-cube generated by the $k$-out model and denoted by $Q^n(k)$. Let $k$ be an integer, $1\leq k \leq n-1$. We let $Q^n(k)$ be the graph that is generated by…

Cyclic codes are an interesting type of linear codes and have applications in communication and storage systems due to their efficient encoding and decoding algorithms. They have been studied for decades and a lot of progress has been made.…

We first prove a one-to-one correspondence between finding Hamiltonian cycles in a cubic planar graphs and finding trees with specific properties in dual graphs. Using this information, we construct an exact algorithm for finding…

Let $G$ be an $n$-vertex graph obtained by adding chords to a cycle of length $n$. Markstr\"{o}m asked for the maximum number of edges in $G$ if there are no two cycles in $G$ with the same length. A simple counting argument shows that such…

If ${\cal H}=(V,{\cal E})$ is a hypergraph, its edge intersection hypergraph $EI({\cal H})=(V,{\cal E}^{EI})$ has the edge set ${\cal E}^{EI}=\{e_1 \cap e_2 \ |\ e_1, e_2 \in {\cal E} \ \wedge \ e_1 \neq e_2 \ \wedge \ |e_1 \cap e_2 |\geq…

Let D be the circulant digraph with n vertices and connection set {2,3,c}. (Assume D is loopless and has outdegree 3.) Work of S.C.Locke and D.Witte implies that if n is a multiple of 6, c is either (n/2) + 2 or (n/2) + 3, and c is even,…

We establish necessary and sufficient conditions for the existence of a decomposition of a complete multigraph into edge-disjoint cycles of specified lengths, or into edge-disjoint cycles of specified lengths and a perfect matching.

A path (cycle) in a $2$-edge-colored multigraph is alternating if no two consecutive edges have the same color. The problem of determining the existence of alternating Hamiltonian paths and cycles in $2$-edge-colored multigraphs is an…

This paper considers the problem of many-to-many disjoint paths in the hypercube $Q_n$ with $f$ faulty vertices and obtains the following result. For any integer $k$ with $1\leq k\leq n-2$, any two sets $S$ and $T$ of $k$ fault-free…

A generalized de Bruijn digraph generalizes a de Bruijn digraph to the case where the number of vertices need not be a pure power of an integer. Hamiltonian cycles in these digraphs thus generalize regular de~Bruijn cycles, and we will thus…

We address various topologies (de Bruijn, chordal ring, generalized Petersen, meshes) in various ways ( isometric embedding, embedding up to scale, embedding up to a distance) in a hypercube or a half-hypercube. Example of obtained…

Disjoint paths are defined as paths between the source and destination nodes where the intermediate nodes in any two paths are disjoint. They are helpful in fault-tolerance routing and securing message distribution in the network. Several…