The UCF Mathematics Colloquium, held every Monday from 3:30pm to 4:30pm in MSB 318, offers a diverse platform for research scholars, faculty, students, and industry experts to share and exchange ideas, fostering discussion and networking across various areas of mathematics.
Dr. Xiaofan Yuan from Arizona State University will speak at this week's colloquium on "Tight Minimum Colored Degree Condition for Rainbow Connectivity."
Abstract: Let $G = (V,E)$ be a graph on $n$ vertices, and let $c : E \to P$, where $P$ is a set of colors. Let $\delta^c(G) = \min_{v \in V} \{ d^{c}(v) \}$ where $d^c(v)$ is the number of colors on edges incident to a vertex $v$ of $G$. In 2011, Fujita and Magnant showed that if $G$ is a graph on $n$ vertices that satisfies $\delta^c(G)\geq n/2$, then for every two vertices $u, v$ there is a properly-colored $u,v$-path in $G$. We show that for sufficiently large graphs $G$ the same bound for $\delta^c(G)$ implies that any two vertices are connected by a rainbow path. This is joint work with Andrzej Czygrinow.
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