► Colorings avoiding disjoint rainbow triangles, tahda queer, Cyrus Young, Wohua Zhou

Advisor:  Juergen Kritschgau
Abstract:  Given an edge-colored graph $G$, we denote the number of colors as $c(G)$, and the number of edges as $e(G)$. An edge-colored graph is rainbow if no two edges share the same color. A proper $mK_3$ is a vertex disjoint union of $m$ rainbow triangles. Rainbow problems have been studied extensively in the context of anti-Ramsey theory, and more recently, in the context of Tur\'{a}n problems. B. Li. et al. European J. Combin. 36 (2014) found that a graph must contain a rainbow triangle if $e(G)+c(G) \geq \binom{n}{2}+ n$. L. Li. and X. Li. Discrete Applied Mathematics 318 (2022) conjectured a lower bound on $e(G)+c(G)$ such that $G$ must contain a proper $mK_3$.

In this paper, we provide a construction that disproves the conjecture, and we introduce results that guarantee the existence of a proper $mK_3$ in general graphs and complete graphs.

► Leaky positive semi-definite zero forcing, Ian Farish, Olivia Elias, Emrys King, Josh Kyei

Advisor:  Ryan Moruzzi
Abstract:  Zero forcing on a graph is an iterative graph coloring process where, starting with an initial set of blue vertices, we try to force other non-blue vertices blue according to a color change rule. The zero forcing number of a graph was first defined in 2008 by a AIM (American Institute of Mathematics) working group as a method of bounding the maximum nullity of a graph. Motivated by monitoring an electrical network, leaks are introduced into the graph hindering the ability of vertices to force. Though we lose the connection with the maximum nullity of a graph, leaks in a graph present interesting new avenues of research pertaining to the area of zero forcing. Our work involved determining the leaky forcing number of graphs for a variation of zero forcing known as positive semidefinite zero forcing.