Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Coloring Drawings of Graphs   We consider face-colorings of drawings of graphs in the plane. Given a
multi-graph $G$ together with a drawing $\Gamma(G)$ in the plane with only
finitely many crossings, we define a face-$k$-coloring of $\Gamma(G)$ to be a
coloring of the maximal connected regions of the drawing, the faces, with $k$
colors such that adjacent faces have different colors. By the $4$-color
theorem, every drawing of a bridgeless graph has a face-$4$-coloring. A drawing
of a graph is facially $2$-colorable if and only if the underlying graph is
Eulerian. We show that every graph without degree 1 vertices admits a
$3$-colorable drawing. This leads to the natural question which graphs $G$ have
the property that each of its drawings has a $3$-coloring. We say that such a
graph $G$ is facially $3$-colorable. We derive several sufficient and necessary
conditions for this property: we show that every $4$-edge-connected graph and
every graph admitting a nowhere-zero $3$-flow is facially $3$-colorable. We
also discuss circumstances under which facial $3$-colorability guarantees the
existence of a nowhere-zero $3$-flow. On the negative side, we present an
infinite family of facially $3$-colorable graphs without a nowhere-zero
$3$-flow. On the positive side, we formulate a conjecture which has a
surprising relation to a famous open problem by Tutte known as the
$3$-flow-conjecture. We prove our conjecture for subcubic and for
$K_{3,3}$-minor-free graphs.

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The categories of this paper are: bayes-an
20
The categories of this paper are: acc-phys
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
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The categories of this paper are: plasm-ph
3
The categories of this paper are: physics
8
The categories of this paper are: hep-ex
5
The categories of this paper are: gr-qc
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The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
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The categories of this paper are: q-fin