相关论文: Hamiltonian Cycles for Finite Weyl Groupoids

Generalizing a result of Conway, Sloane, and Wilkes for real reflection groups, we show the Cayley graph of an imprimitive complex reflection group with respect to standard generating reflections has a Hamiltonian cycle. This is consistent…

A Hamilton cycle in a graph $\Gamma$ is a cycle passing through every vertex of $\Gamma$. A Hamiltonian decomposition of $\Gamma$ is a partition of its edge set into disjoint Hamilton cycles. One of the oldest results in graph theory is…

We prove Wise's $W$-cycles conjecture. Consider a compact graph $\Gamma'$ immering into another graph $\Gamma$. For any immersed cycle $\Lambda:S^1\to \Gamma$, we consider the map $\Lambda'$ from the circular components $\mathbb{S}$ of the…

Let \( G \) be a finite non-cyclic group. Define \( \mathrm{Cyc}(G) \) as the set of all elements \( a \in G \) such that for any $b\in G$, the subgroup \( \langle a, b \rangle \) is cyclic. The \emph{non-cyclic graph} $\Gamma(G)$ of \( G…

We show that if G is a finite group whose commutator subgroup [G,G] has order 2p, where p is an odd prime, then every connected Cayley graph on G has a hamiltonian cycle.

We show that if G is any nilpotent, finite group, and the commutator subgroup of G is cyclic, then every connected Cayley graph on G has a hamiltonian cycle.

We study Hamilton circuits of the Cayley graphs of the Weyl groupoids of the generalized quantum groups, or the quantum double of the Nichols algebras of diagonal-type, with finite root systems. We prove the existence of a Hamilton circuit…

Motivated by the Gray code interpretation of Hamiltonian cycles in Cayley graphs, we investigate the existence of Hamiltonian cycles in tope graphs of hyperplane arrangements, with a focus on simplicial, reflection, and supersolvable…

We initiate the study of Hamiltonian cycles up to symmetries of the underlying graph. Our focus lies on the extremal case of Hamiltonian-transitive graphs, i.e., Hamiltonian graphs where, for every pair of Hamiltonian cycles, there is a…

A graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex-transitive graphs with finitely many ends, and also discuss…

Let $G$ be a finite, non-abelian group of the form $G = A N$, where $A \leq G$ is abelian, and $N \trianglelefteq G$ is cyclic. We prove that the commuting graph $\Gamma(G)$ of $G$ is either a connected graph of diameter at most four, or…

We study the derangement graph $\Gamma_n$ whose vertex set consists of all permutations of $\{1,\ldots,n\}$, where two vertices are adjacent if and only if their corresponding permutations differ at every position. It is well-known that…

Finding general conditions which ensure that a graph is Hamiltonian is a central topic in graph theory. An old and well known conjecture in the area states that any $d$-regular $n$-vertex graph $G$ whose second largest eigenvalue in…

A hamiltonian cycle system (HCS, for short) of a graph $\Gamma$ is a partition of the edges of $\Gamma$ into hamiltonian cycles. A HCS is cyclic when it is invariant under a cyclic permutation of all the vertices of $\Gamma$; the existence…

Let ${\mathcal W}_n$ denote any of the three families of classical Weyl groups: the symmetric groups ${\mathcal S}_n$, the hyperoctahedral groups (signed permutation groups) ${\mathcal S}^B_n$, or the even-signed permutation groups…

In this article we discuss the question of presence of Hamiltonian cycle in the un-directed power graph of a group. In the process we develop weighted Hamiltonian cycle concept and prove a few general results regarding the Hamiltonian…

Let $W$ be a Coxeter group whose proper parabolic subgroups are finite. According to Theorem~1.12 of [1], if the module of a finite $W$-digraph $\Gamma$ is isomorphic to the module of a $W$-graph over $Q$, then $\Gamma$ is acyclic. We…

For a graph $G$, the $t$-th power $G^t$ is the graph on $V(G)$ such that two vertices are adjacent if and only if they have distance at most $t$ in $G$; and the $t$-th bi-power $G_B^t$ is the graph on $V(G)$ such that two vertices are…

Assume G is a finite group, such that |G|= 6pq or 7pq, where p and q are distinct prime numbers, and let S be a generating set of G. We prove there is a Hamiltonian cycle in the corresponding Cayley graph Cay(G;S).

Let $S_n$ and $A_{n}$ denote the symmetric and alternating group on the set $\{1,.., n\},$ respectively. In this paper we are interested in the second largest eigenvalue $\lambda_{2}(\Gamma)$ of the Cayley graph $\Gamma=Cay(G,H)$ over…