#### Hall 123

#### Embedding of metric graphs on hyperbolic surfaces

#### Bidyut Sanki

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*An embedding of a metric graph $(G, d)$ into a closed hyperbolic surface is called \emph{essential} if it is isometric and each complementary region has a negative Euler characteristic. We show, by construction, that every metric graph can be essentially embedded (up to scaling) on a closed hyperbolic surface. The essential genus $g_e(G)$ of a metric graph $(G, d)$ is defined as the lowest integer among the genera of the surfaces on which the metric graph can be essentially embedded. In the next result we establish a formula to compute $g_e(G)$. Furthermore, we show that for every integer $g\geq g_e(G)$, $(G, d)$ can be essentially embedded up to scaling on a surface of genus $g$.*

Next, we study minimal embedding, where each complementary region has Euler characteristic $-1$. We define a parameter $g_e^{\max}(G)$ to be the maximal essential genus of a graph $(G, d)$ in minimal embedding. We compute an upper bound for $g_e^{\max}(G)$ and prove that for any integer $g$, $g_e(G)\leq g \leq g_e^{\max}(G)$, there exists a closed surface on which $(G, d)$ can be minimally embedded. Finally, we describe a method explicitly for an essential embedding of $(G, d)$ where $g_e(G)$ and $g_e^{\max}(G)$ are realized

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