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{\bf Andr\'e K\"undgen, Eric Mendelsohn, Vitaly Voloshin}
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{\bf Colouring Planar Mixed Hypergraphs}
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A mixed hypergraph is a triple ${\cal H} = (V,{\cal C}, {\cal D})\;$
where $V$ is the vertex set and ${\cal C}$ and ${\cal D}$ are families
of subsets of $V$, the ${\cal C}$-edges and ${\cal D}$-edges,
respectively. A $k$-colouring of ${\cal H}$ is a mapping
$c: V\rightarrow [k]$ such that each ${\cal C}$-edge has
at least two vertices with a ${\cal C}$ommon colour and each ${\cal D}$-edge
has at least two vertices of ${\cal D}$ifferent colours.
${\cal H}$ is called a planar mixed hypergraph if its bipartite
representation is a planar graph.
Classic graphs are the special case of mixed
hypergraphs when ${\cal C}=\emptyset$ and all the ${\cal D}$-edges have
size 2, whereas in a bi-hypergraph ${\cal C} = {\cal D}$.
We investigate the colouring properties of planar mixed hypergraphs.
Specifically, we show that maximal planar bi-hypergraphs are
2-colourable, find formulas for their chromatic polynomial and
chromatic spectrum in terms of 2-factors in the dual,
prove that their chromatic spectrum is gap-free and provide
a sharp estimate on the maximum number of colours in a colouring.
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