arXiv subjects

Thomas Tucker

Publications and source records attributed to Thomas Tucker.

5 recordsLinked to original sources

arXiv2020

On the genera of polyhedral embeddings of cubic graph

In this article we present theoretical and computational results on the existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs. We also describe an efficient algorithm to compute all polyhedral embeddings of a given cubic graph and constructions for cubic graphs with some special properties of their polyhedral embeddings. Some key results are that even cubic graphs with a polyhedral embedding on the torus can also have polyhedral embeddings in arbitrarily high genus, in fact in a genus {\em close} to the theoretical maximum for that number of vertices, and that there is no bound on the number of genera in which a cubic graph can have a polyhedral embedding. While these results suggest a large variety of polyhedral embeddings, computations for up to 28 vertices suggest that by far most of the cubic graphs do not have a polyhedral embedding in any genus and that the ratio of these graphs is increasing with the number of vertices.

math.CO↗

arXiv2018

Virtual replicas of real places: Experimental investigations

The emergence of social virtual reality (VR) experiences, such as Facebook Spaces, Oculus Rooms, and Oculus Venues, will generate increased interest from users who want to share real places (both personal and public) with their fellow users in VR. At the same time, advances in scanning and reconstruction technology are making the realistic capture of real places more and more feasible. These complementary pressures mean that the representation of real places in virtual reality will be an increasingly common use case for VR. Despite this, there has been very little research into how users perceive such replicated spaces. This paper reports the results from a series of three user studies investigating this topic. Taken together, these results show that getting the scale of the space correct is the most important factor for generating a "feeling of reality", that it is important to avoid incoherent behaviors (such as floating objects), and that lighting makes little difference to perceptual similarity.

cs.HC↗

arXiv2017

Distinguishing graphs of maximum valence 3

The distinguishing number $D(G)$ of a graph $G$ is the smallest number of colors that is needed to color $G$ such that the only color preserving automorphism is the identity. We give a complete classification for all connected graphs $G$ of maximum valence $\triangle(G)=3$ and distinguishing number $D(G) = 3$. As one of the consequences we get that all infinite connected graphs with $\triangle(G)=3$ are 2-distinguishable.

math.CO↗

arXiv2017

ABC implies a Zsigmondy principle for ramification

Let $K$ be a number field or a function field of characteristic 0. If $K$ is a number field, assume the $abc$-conjecture for $K$. We prove a variant of Zsigmondy's theorem for ramified primes in preimage fields of rational functions in $K(x)$ that are not postcritically finite. For example, suppose $K$ is a number field and $f\in K[x]$ is not postcritically finite, and let $K_n$ be the field generated by the $n$th iterated preimages under $f$ of $β\in K$. We show that for all large $n$, there is a prime of $K$ that ramifies in $K_n$ and does not ramify in $K_m$ for any $m<n$.

math.NT↗

arXiv2011

Preperiodic points for families of polynomials

Let $a(λ)$ and $b(λ)$ be two polynomials with coefficients in complex numbers and let $f_{\lamb$ be a one-parameter family of polynomials indexed by all complex numbers $λ$. We study whether there exist infinitely many complex numbers $λ$ such that both $a(λ)$ and $b(λ)$ are preperiodic for $f_λ$.

math.DS↗