Hudson LaFayette | Searcharxiv

arXiv2023

Bootstrap Percolation, Connectivity, and Graph Distance

Bootstrap Percolation is a process defined on a graph which begins with an initial set of infected vertices. In each subsequent round, an uninfected vertex becomes infected if it is adjacent to at least $r$ previously infected vertices. If an initially infected set of vertices, $A_0$, begins a process in which every vertex of the graph eventually becomes infected, then we say that $A_0$ percolates. In this paper we investigate bootstrap percolation as it relates to graph distance and connectivity. We find a sufficient condition for the existence of cardinality 2 percolating sets in diameter 2 graphs when $r = 2$. We also investigate connections between connectivity and bootstrap percolation and lower and upper bounds on the number of rounds to percolation in terms of invariants related to graph distance.

math.CO↗

arXiv2022

The $t$-Tone Chromatic Number of Classes of Sparse Graphs

For a graph $G$ and $t,k\in\mathbb{Z}^+$ a \emph{$t$-tone $k$-coloring} of $G$ is a function $f:V(G)\rightarrow \binom{[k]}{t}$ such that $|f(v)\cap f(w)| < d(v,w)$ for all distinct $v,w \in V(G)$. The \emph{$t$-tone chromatic number} of $G$, denoted $τ_t(G)$, is the minimum $k$ such that $G$ is $t$-tone $k$-colorable. For small values of $t$, we prove sharp or nearly sharp upper bounds on the $t$-tone chromatic number of various classes of sparse graphs. In particular, we determine $τ_2(G)$ exactly when $\textrm{mad}(G) < 12/5$ and bound $τ_2(G)$, up to a small additive constant, when $G$ is outerplanar. We also determine $τ_t(C_n)$ exactly when $t\in\{3,4,5\}$.

math.CO↗