Searcharxiv⌕ Search

arXiv subjects

Dave Witte Morris

Publications and source records attributed to Dave Witte Morris.

At least 19 recordsLinked to original sources
arXiv2026

Isomorphisms of abelian Cayley graphs with their natural edge-colouring

We prove that if $\varphi$ is an isomorphism between two connected Cayley graphs of abelian groups, and $\varphi$ respects the natural edge-colourings of the Cayley graphs, then $\varphi$ is the composition of a group isomorphism and a colour-preserving graph automorphism. This implies that if every colour-preserving automorphism of a connected abelian Cayley graph $Cay(G;S)$ is an affine map, then the same is true for every colour-permuting automorphism. We also show that this property holds if and only if the subgroup generated by $\{\, s \in S \mid 2s \neq c \,\} \cup \{c\}$ has index $\le 2$ for every element $c$ of order $2$ in $G$.

math.CO↗
arXiv2026

Composing group automorphisms with colour-preserving automorphisms of Cayley graphs

We show that if $\varphi$ is a colour-permuting automorphism of a connected, finite Cayley graph, and the order of the Cayley graph is either odd or square-free, then $\varphi$ is the composition of a group automorphism and a colour-preserving graph automorphism. Some analogous results are also established for isomorphisms between two different Cayley graphs.

math.CO↗
arXiv2024

Non-left-orderability of lattices in higher-rank semisimple Lie groups (after Deroin and Hurtado)

Let $G$ be a connected, semisimple, real Lie group with finite centre, with real rank at least two. B.Deroin and S.Hurtado recently proved the 30-year-old conjecture that no irreducible lattice in $G$ has a left-invariant total order. (Equivalently, they proved that no such lattice has a nontrivial, orientation-preserving action on the real line.) We will explain many of the main ideas of the proof, by using them to prove the analogous result for lattices in $p$-adic semisimple groups. The $p$-adic case is easier, because some of the technical issues do not arise.

math.GR↗
arXiv2024

Colour-permuting automorphisms of complete Cayley graphs

Let $G$ be a (finite or infinite) group, and let $K_G = \mathrm{Cay}(G;G \smallsetminus \{1\} )$ be the complete graph with vertex set $G$, considered as a Cayley graph of $G$. Being a Cayley graph, it has a natural edge-colouring by sets of the form $\{s, s^{-1}\}$ for $s \in G$. We prove that every colour-permuting automorphism of $K_G$ is an affine map, unless $G \cong Q_8 \times B$, where $Q_8$ is the quaternion group of order $8$, and $B$ is an abelian group, such that $b^2$ is trivial for all $b \in B$. We also prove (without any restriction on $G$) that every colour-permuting automorphism of $K_G$ is the composition of a group automorphism and a colour-preserving graph automorphism. This was conjectured by D. P. Byrne, M. J. Donner, and T. Q. Sibley in 2013.

math.CO↗
arXiv2023

On vertex-transitive graphs with a unique hamiltonian cycle

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 some examples with infinitely many ends. In particular, we show each nonabelian free group $F_n$ has a Cayley graph of degree $2n + 2$ that has a unique hamiltonian circle. (A weaker statement had been conjectured by A. Georgakopoulos.) Furthermore, we prove that these Cayley graphs of $F_n$ are outerplanar.

math.CO↗
arXiv2023

Cayley graphs of order 8pq are hamiltonian

We give a computer-assisted proof that if $G$ is a finite group of order $8pq$, where $p$ and $q$ are distinct primes, then every connected Cayley graph on $G$ has a hamiltonian cycle.

math.CO↗
arXiv2023

Cayley graphs of order kp are hamiltonian for k < 48

We provide a computer-assisted proof that if G is any finite group of order kp, where k < 48 and p is prime, then every connected Cayley graph on G is hamiltonian (unless kp = 2). As part of the proof, it is verified that every connected Cayley graph of order less than 48 is either hamiltonian connected or hamiltonian laceable (or has valence less than three).

math.CO↗
arXiv2023

Non-Cayley-Isomorphic Cayley graphs from non-Cayley-Isomorphic Cayley digraphs

A finite group $G$ is a "non-DCI group" if there exist subsets $S_1$ and $S_2$ of $G$, such that the associated Cayley digraphs $C\overrightarrow{ay}(G;S_1)$ and $C\overrightarrow{ay}(G;S_2)$ are isomorphic, but no automorphism of $G$ carries $S_1$ to $S_2$. Furthermore, $G$ is a "non-CI group" if the subsets $S_1$ and $S_2$ can be chosen to be closed under inverses, so we have undirected Cayley graphs $Cay(G;S_1)$ and $Cay(G;S_2)$. We show that if $p$ is a prime number, and the elementary abelian $p$-group $(\mathbb{Z}_p)^r$ is a non-DCI group, then $(\mathbb{Z}_p)^{r+3}$ is a non-CI group. In most cases, we can also show that $(\mathbb{Z}_p)^{r+2}$ is a non-CI group. In particular, from Pablo Spiga's proof that $(\mathbb{Z}_p)^8$ is a non-DCI group, we conclude that $(\mathbb{Z}_3)^{10}$ is a non-CI group. This is the first example of a non-CI elementary abelian $3$-group.

math.CO↗
arXiv2023

Automorphisms of the canonical double cover of a toroidal grid

The Cartesian product of two cycles (of length m and length n) has a natural embedding on the torus, such that each face of the embedding is a 4-cycle. The toroidal grid Qd(m,n,r) is a generalization of this in which there is a shift by r when traversing the meridian of length m. In 2008, Steve Wilson found two interesting infinite families of (nonbipartite) toroidal grids that are unstable. (By definition, this means that the canonical bipartite double cover of the grid has more than twice as many automorphisms as the grid has.) It is easy to see that bipartite grids are also unstable, because the canonical double cover is disconnected. Furthermore, there are degenerate cases in which there exist two different vertices that have the same neighbours. This paper proves Wilson's conjecture that Qd(m,n,r) is stable for all other values of the parameters. In addition, we prove an analogous conjecture of Wilson for the triangular grids Tr(m,n,r) that are obtained by adding a diagonal to each face of Qd(m,n,r) (with all of the added diagonals parallel to each other).

math.CO↗
arXiv2022

Hamiltonicity after reversing the directed edges at a vertex of a Cartesian product

Let $\vec C_m$ and $\vec C_n$ be directed cycles of length $m$ and $n$, with $m,n \ge 3$, and let $P(\vec C_m \mathbin{\Box} \vec C_n)$ be the digraph that is obtained from the Cartesian product $\vec C_m \mathbin{\Box} \vec C_n$ by choosing a vertex $v$, and reversing the orientation of all four directed edges that are incident with $v$. (This operation is called "pushing" at the vertex $v$.) By applying a special case of unpublished work of S.X.Wu, we find elementary number-theoretic necessary and sufficient conditions for the existence of a hamiltonian cycle in $P(\vec C_m \mathbin{\Box} \vec C_n)$. A consequence is that if $P(\vec C_m \mathbin{\Box} \vec C_n)$ is hamiltonian, then $\gcd(m,n) = 1$, which implies that $\vec C_m \mathbin{\Box} \vec C_n$ is not hamiltonian. This final conclusion verifies a conjecture of J.B.Klerlein and E.C.Carr.

math.CO↗
arXiv2022

Arc-disjoint hamiltonian paths in Cartesian products of directed cycles

We show that if $C_1$ and $C_2$ are directed cycles (of length at least two), then the Cartesian product $C_1 \Box C_2$ has two arc-disjoint hamiltonian paths. (This answers a question asked by J. A. Gallian in 1985.) The same conclusion also holds for the Cartesian product of any four or more directed cycles (of length at least two), but some cases remain open for the Cartesian product of three directed cycles. We also discuss the existence of arc-disjoint hamiltonian paths in $2$-generated Cayley digraphs on (finite or infinite) abelian groups.

math.CO↗
arXiv2021

On automorphisms of the double cover of a circulant graph

A graph $X$ is said to be "unstable" if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is "nontrivially unstable" if it is unstable, connected, and nonbipartite, and no two distinct vertices of X have exactly the same neighbors. We find three new conditions that each imply a circulant graph is unstable. (These yield infinite families of nontrivially unstable circulant graphs that were not previously known.) We also find all of the nontrivially unstable circulant graphs of order $2p$, where $p$ is any prime number. Our results imply that there does not exist a nontrivially unstable circulant graph of order $n$ if and only if either $n$ is odd, or $n < 8$, or $n = 2p$, for some prime number $p$ that is congruent to $3$ modulo $4$.

math.CO↗
arXiv2021

Automorphisms of the double cover of a circulant graph of valency at most 7

A graph $X$ is said to be unstable if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is nontrivially unstable if it is unstable, connected, and non-bipartite, and no two distinct vertices of X have exactly the same neighbors. We find all of the nontrivially unstable circulant graphs of valency at most $7$. (They come in several infinite families.) We also show that the instability of each of these graphs is explained by theorems of Steve Wilson. This is best possible, because there is a nontrivially unstable circulant graph of valency $8$ that does not satisfy the hypotheses of any of Wilson's four instability theorems for circulant graphs.

math.CO↗
arXiv2020

Stability of Cayley graphs on abelian groups of odd order

Let $X$ be a connected Cayley graph on an abelian group of odd order, such that no two distinct vertices of $X$ have exactly the same neighbours. We show that the direct product $X \times K_2$ (also called the "canonical double cover" of $X$) has only the obvious automorphisms (namely, the ones that come from automorphisms of its factors $X$ and $K_2$). This means that $X$ is "stable". The proof is short and elementary. The theory of direct products implies that $K_2$ can be replaced with members of a much more general family of connected graphs.

math.CO↗
arXiv2020

Groups for which it is easy to detect graphical regular representations

We say that a finite group G is "DRR-detecting" if, for every subset S of G, either the Cayley digraph Cay(G,S) is a digraphical regular representation (that is, its automorphism group acts regularly on its vertex set) or there is a nontrivial group automorphism phi of G such that phi(S) = S. We show that every nilpotent DRR-detecting group is a p-group, but that the wreath product of two cyclic groups of order p is not DRR-detecting, for every odd prime p. We also show that if G and H are nontrivial groups that admit a digraphical regular representation and either gcd(|G|,|H|) = 1, or H is not DRR-detecting, then the direct product G x H is not DRR-detecting. Some of these results also have analogues for graphical regular representations.

math.CO↗
arXiv2020

Quasi-Isometric Bounded Generation by ${\mathbb Q}$-Rank-One Subgroups

We say that a subset $X$ quasi-isometrically boundedly generates a finitely generated group $Γ$ if each element $γ$ of a finite-index subgroup of $Γ$ can be written as a product $γ= x_1 x_2 \cdots x_r$ of a bounded number of elements of $X$, such that the word length of each $x_i$ is bounded by a constant times the word length of $γ$. A. Lubotzky, S. Mozes, and M.S. Raghunathan observed in 1993 that ${\rm SL}(n,{\mathbb Z})$ is quasi-isometrically boundedly generated by the elements of its natural ${\rm SL}(2,{\mathbb Z})$ subgroups. We generalize (a slightly weakened version of) this by showing that every $S$-arithmetic subgroup of an isotropic, almost-simple ${\mathbb Q}$-group is quasi-isometrically boundedly generated by standard ${\mathbb Q}$-rank-1 subgroups.

math.GR↗
arXiv2017

Relative Property (T) for Nilpotent Subgroups

We show that relative Property (T) for the abelianization of a nilpotent normal subgroup implies relative Property (T) for the subgroup itself. This and other results are a consequence of a theorem of independent interest, which states that if $H$ is a closed subgroup of a locally compact group $G$, and $A$ is a closed subgroup of the center of $H$, such that $A$ is normal in $G$, and $(G/A, H/A)$ has relative Property (T), then $(G, H^{(1)})$ has relative Property (T), where $H^{(1)}$ is the closure of the commutator subgroup of $H$. In fact, the assumption that $A$ is in the center of $H$ can be replaced with the weaker assumption that $A$ is abelian and every $H$-invariant finite measure on the unitary dual of $A$ is supported on the set of fixed points.

math.RT↗