# Complete minors in orientable surfaces

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Gašper Fijavž

IMFM and University of Ljubljana

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**Minisymposium:**
GRAPH IMBEDDINGS AND MAP SYMMETRIES

**Content:**
Assume that a graph $G$ embeds in a surface $\Sigma$. A theorem of Robertson and Seymour states that there exists a constant $c_\Sigma$, so that every graph which embeds in $\Sigma$ with face-width $\ge c_\Sigma$ contains $G$ as a minor.
Let $\Sigma$ be an arbitrary \emph{orientable} surface of genus $\ge 1$.
We prove that there exists an absolute constant $c$ (independent on the choice of $\Sigma$),
so that:
\begin{itemize}
\item if $G$ embeds in $\Sigma$ with face-width $\ge c$, then $G$ contains a $K_7$ minor, and
\item if $G$ is not bipartite and embeds in $\Sigma$ with face-width $\ge c$ while having all faces of even length, then $G$ contains an \emph{odd} $K_5$ minor.
\end{itemize}
This is in part joint work with Atsuhiro Nakamoto.