Related papers: Matchings in Hypercubes Extend to Long Cycles
A bond in a graph is a minimal nonempty edge-cut. A connected graph $G$ is dual Hamiltonian if the vertex set can be partitioned into two subsets $X$ and $Y$ such that the subgraphs induced by $X$ and $Y$ are both trees. There is much…
In this paper, we consider a random geometric graph (RGG)~\(G\) on~\(n\) nodes with adjacency distance~\(r_n\) just below the Hamiltonicity threshold and construct Hamiltonian cycles using additional edges called bridges. The bridges by…
We study the existence of plane substructures in drawings of the $d$-dimensional hypercube graph $Q_d$. We construct drawings of $Q_d$ which contain no plane subgraph with more than $2d-2$ edges, no plane path with more than $2d-3$ edges,…
It is proved that for $n \geq 6$, the number of perfect matchings in a simple connected cubic graph on $2n$ vertices is at most $4 f_{n-1}$, with $f_n$ being the $n$-th Fibonacci number. The unique extremal graph is characterized as well.…
In a sequence of four papers, we prove the following results (via a unified approach) for all sufficiently large $n$: (i) [1-factorization conjecture] Suppose that $n$ is even and $D\geq 2\lceil n/4\rceil -1$. Then every $D$-regular graph…
Let $Q^d$ be the $d$-dimensional binary hypercube. We form a random subgraph $Q^d_p\subseteq Q^d$ by retaining each edge of $Q^d$ independently with probability $p$. We show that, for every constant $\varepsilon>0$, there exists a constant…
Twisted hypercubes are generalizations of the Boolean hypercube, obtained by iteratively connecting two instances of a graph by a uniformly random perfect matching. Dudek et al. showed that when the two instances are independent, these…
We investigate the existence of a rainbow Hamilton cycle in a uniformly edge-coloured randomly perturbed digraph. We show that for every $\delta \in (0,1)$ there exists $C = C(\delta) > 0$ such that the following holds. Let $D_0$ be an…
One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erdos-Renyi random graph G_{n,p} is around p ~ (log n + log log n) / n. Much research has been done to extend this to…
We introduce and study a $d$-dimensional generalization of Hamiltonian cycles in graphs - the Hamiltonian $d$-cycles in $K_n^d$ (the complete simplicial $d$-complex over a vertex set of size $n$). Those are the simple $d$-cycles of a…
A tight Hamilton cycle in a $k$-uniform hypergraph ($k$-graph) $G$ is a cyclic ordering of the vertices of $G$ such that every set of $k$ consecutive vertices in the ordering forms an edge. R\"{o}dl, Ruci\'{n}ski, and Szemer\'{e}di proved…
We define and study a special type of hypergraph. A $\sigma$-hypergraph $H= H(n,r,q$ $\mid$ $\sigma$), where $\sigma$ is a partition of $r$, is an $r$-uniform hypergraph having $nq$ vertices partitioned into $ n$ classes of $q$ vertices…
We construct an explicit Hamiltonian cycle in the state graph of the 5-puzzle on a toroidal 2x 3 grid, a graph with 720 vertices. The cycle is described by a short symbolic sequence of 48 moves over the alphabet {L,R,V}, repeated $15$…
The CHY construction naturally associates a vector in $\mathbb{R}^{(n-3)!}$ to every 2-regular graph with $n$ vertices. Partial amplitudes in the biadjoint scalar theory are given by the inner product of vectors associated with a pair of…
The Hamiltonian cycle polynomial can be evaluated to count the number of Hamiltonian cycles in a graph. It can also be viewed as a list of all spanning cycles of length $n$. We adopt the latter perspective and present a pair of original…
For integers $k\geq 1$ and $n\geq 2k+1$ the Kneser graph $K(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ and an edge between any two vertices (=sets) that are disjoint. The bipartite Kneser graph $H(n,k)$ has as…
Define the middle layer graph as the graph whose vertex set consists of all bitstrings of length $2n+1$ that have exactly $n$ or $n+1$ entries equal to 1, with an edge between any two vertices for which the corresponding bitstrings differ…
The subdivided double construction on 4-regular graphs was used by Poto\v{c}nik and Wilson to explore semi-symmetric (edge-transitive but not vertex-transitive) graphs, and can be used to construct every semi-symmetric 4-regular graph that…
In this short note, we prove that for \beta < 1/5 every graph G with n vertices and n^{2-\beta} edges contains a subgraph G' with at least cn^{2-2\beta} edges such that every pair of edges in G' lie together on a cycle of length at most 8.…
An $n$-vertex graph is Hamiltonian if it contains a cycle that covers all of its vertices, and it is pancyclic if it contains cycles of all lengths from $3$ up to $n$. In 1972, Erd\H{o}s conjectured that every Hamiltonian graph with…