Infinite Families of Asymmetric Graphs

A graph $G$ is \textit{asymmetric} if its automorphism group of vertices is trivial. Asymmetric graphs were introduced by Erd\H{o}s and R\'{e}nyi in 1963. They showed that the probability of a graph on $n$ vertices being asymmetric tends to $1$ as $n$ tends to infinity. In this paper, we first give consider the number of asymmetric trees, a question posed by Erd\H{o}s and R\'enyi. We give a partial result, showing that the number of asymmetric subdivided stars is approximately $q(n-1) - \lfloor \frac{n-1}{2} \rfloor$ where $q(n)$ is the number of ways to sum to $n$ using distinct positive integers, found by Hardy and Ramanujan in 1918. We also investigate cubic Hamiltonian graphs where asymmetry, at least for small values of $n$, seems to be rare. It is known that none of the cubic Hamiltonian graphs on $4\leq n\leq 10$ vertices are asymmetric, and of the $80$ cubic Hamiltonian graphs on $12$ vertices only $5$ are asymmetric. We give a construction of an infinite family of cubic Hamiltonian graphs that are asymmetric. Then we present an infinite family of quartic Hamiltonian graphs that are asymmetric. We use both of the above results for cubic and quartic asymmetric Hamiltonian graphs to establish the existence of $k$-regular asymmetric Hamiltonian graphs for all $k\geq 3$.