Related papers: Graph Universal Cycles: Compression and Connection…
We study universal cycles of the set ${\cal P}(n,k)$ of $k$-partitions of the set $[n]:=\{1,2,\ldots,n\}$ and prove that the transition digraph associated with ${\cal P}(n,k)$ is Eulerian. But this does not imply that universal cycles (or…
We prove some functional equations involving the (classical) matching polynomials of path and cycle graphs and the $d$-matching polynomial of a cycle graph. A matching in a (finite) graph $G$ is a subset of edges no two of which share a…
We say that a $k$-uniform hypergraph $C$ is a Hamilton cycle of type $\ell$, for some $1\le \ell \le k$, if there exists a cyclic ordering of the vertices of $C$ such that every edge consists of $k$ consecutive vertices and for every pair…
We study the Laplacian of the undirected De Bruijn graph over an alphabet $A$ of order $k$. While the eigenvalues of this Laplacian were found in 1998 by Delorme and Tillich [1], an explicit description of its eigenvectors has remained…
We define generalized de Bruijn words as those words having a Burrows-Wheeler transform that is a concatenation of permutations of the alphabet. We show that generalized de Bruijn words are in 1-to-1 correspondence with Hamiltonian cycles…
The triangle graph of a graph $G$, denoted by ${\cal T}(G)$, is the graph whose vertices represent the triangles ($K_3$ subgraphs) of $G$, and two vertices of ${\cal T}(G)$ are adjacent if and only if the corresponding triangles share an…
A supergrid graph is a finite induced subgraph of the infinite graph associated with the two-dimensional supergrid. The supergrid graphs contain grid graphs and triangular grid graphs as subgraphs. The Hamiltonian cycle problem for grid and…
Global control offers a promising route to scalable quantum computing. A recent conjecture by Hu et al. (arXiv:2508.19075) proposes that any connected qubit graph equipped with global Ising-type interactions and tunable global transverse…
A biased graph consists of a graph $G$ together with a collection of distinguished cycles of $G$, called balanced cycles, with the property that no theta subgraph contains exactly two balanced cycles. Perhaps the most natural biased graphs…
Breakpoint graphs are ubiquitous structures in the field of genome rearrangements. Their cycle decomposition has proved useful in computing and bounding many measures of (dis)similarity between genomes, and studying the distribution of…
We introduce the concept of complete edge-colored permutation graphs as complete graphs that are the edge-disjoint union of "classical" permutation graphs. We show that a graph $G=(V,E)$ is a complete edge-colored permutation graph if and…
The cycle set of a graph $G$ is the set consisting of all sizes of cycles in $G$. Answering a conjecture of Erd\H{o}s and Faudree, Verstra\"{e}te showed that there are at most $2^{n - n^{1/10}}$ different cycle sets of graphs with $n$…
In card-based cryptography, a deck of physical cards is used to achieve secure computation. A shuffle, which randomly permutes a card-sequence along with some probability distribution, ensures the security of a card-based protocol. The…
In the literature, the notion of discrepancy is used in several contexts, even in the theory of graphs. Here, for a graph $G$, $\{-1, 1\}$ labels are assigned to the edges, and we consider a family $\mathcal{S}_G$ of (spanning) subgraphs of…
The Bubble-sort graph $BS_n,\,n\geqslant 2$, is a Cayley graph over the symmetric group $Sym_n$ generated by transpositions from the set $\{(1 2), (2 3),\ldots, (n-1 n)\}$. It is a bipartite graph containing all even cycles of length…
For $0\leq \ell <k$, a Hamiltonian $\ell$-cycle in a $k$-uniform hypergraph $H$ is a cyclic ordering of the vertices of $H$ in which the edges are segments of length $k$ and every two consecutive edges overlap in exactly $\ell$ vertices. We…
Median graphs are connected graphs in which for all three vertices there is a unique vertex that belongs to shortest paths between each pair of these three vertices. In this paper we provide several novel characterizations of planar median…
In 2021, Gupta and Suzumura proposed a novel algorithm for enumerating all bounded-length simple cycles in directed graphs. In this work, we present concrete examples demonstrating that the proposed algorithm fails to enumerate certain…
Cycle polytopes of matroids have been introduced in combinatorial optimization as a generalization of important classes of polyhedral objects like cut polytopes and Eulerian subgraph polytopes associated to graphs. Here we start an…
This paper presents a general and systematic discussion of various symbolic representations of iterated maps through subshifts. We give a unified model for all continuous maps on a metric space, by representing a map through a general…