Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Simple and Direct Proof of MacLane's Planarity Criterion   We give a simple proof of MacLane's algebraic planarity criterion for graphs.
This proof does not use any other known planarity criteria.
  Keywords: graph, planarity, cycle space, a simple basis of a graph.

5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
26
The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th