Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Chords of 2-factors in planar cubic bridgeless graphs   We show that every edge in a 2-edge-connected planar cubic graph is either
contained in a 2-edge-cut or is a chord of some cycle that is contained in a
2-factor of the graph. As a consequence, we show that every edge in a
cyclically 4-edge-connected planar cubic graph, except $K_2^3$ and $K_4$, is
contained in a perfect matching whose removal disconnects the graph. We obtain
a complete characterization of 2-edge-connected planar cubic graphs that have
an edge such that every 2-factor containing the edge is a Hamiltonian cycle,
and also of those that have an edge such that the complement of every perfect
matching containing the edge is a Hamiltonian cycle. Another immediate
consequence of the main result is that for any two edges contained in a facial
cycle of a 2-edge-connected planar cubic graph, there exists a 2-factor in the
graph such that both edges are contained in the same cycle of the 2-factor. We
conjecture that this property holds for any two edges in a 2-edge-connected
planar cubic graph, and prove it for planar cubic bipartite graphs. The main
result is proved in the dual form by showing that every plane triangulation
admits a vertex 3-coloring such that no face is monochromatic and there is
exactly one specified edge between a specified pair of color classes.

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The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
26
The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
16
The categories of this paper are: stat
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The categories of this paper are: atom-ph
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg