arXiv subjects
Publications and source records attributed to Elyssa Stephens.
In this paper, we show that shadowing and inverse shadowing are generic properties for spaces in which there exists an $\varepsilon$-retraction onto a graph for any $\varepsilon>0$.
In this paper, we explore chromatic numbers subject to various local modular constraints. For fixed $n$, we consider proper integer colorings of a graph $G$ for which the closed and open neighborhood sums have nonzero remainders modulo $n$ and provide bounds for the associated chromatic numbers $\chi_n(G)$ and $\chi_{(n)}(G)$, respectively. In addition, we provide bounds for $\chi_{(n,k)}(G)$, the minimal order of a proper integer coloring of $G$ with open neighborhood sums congruent to $k\mod n$ (when such a coloring exists) as well as precise values for certain families of graphs.
We examine dynamical systems with the property that pseudo-orbits can be traced by small diameter sets with bounded cardinality. In particular, we show that mixing sofic subshifts and surjective dynamical systems with the specification property have this property, and in these systems, it is sufficient to consider small sets of cardinality no more than two. We also prove that a more general class of subshifts, the subshifts of quasi-finite type, exhibit this property.
Generalizing the notion of odd-sum colorings, a $\mathbb{Z}$-labeling of a graph $G$ is called a closed coloring with remainder $k\mod n$ if the closed neighborhood label sum of each vertex is congruent to $k\mod n$. If such colorings exist, we write $\chi_{n,k}(G)$ for the minimum number of colors used for a closed coloring with remainder $k\mod n$ such that no neighboring vertices have the same color. General estimates for $\chi_{n,k}(G)$ are given along with evaluations of $\chi_{n,k}(G)$ for some finite and infinite order graphs.