Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Wheels in planar graphs and Haj\'os graphs   It was conjectured by Haj\'{o}s that graphs containing no $K_5$-subdivision
are 4-colorable. Previous results show that any possible minimum counterexample
to Haj\'{o}s' conjecture, called Haj\'{o}s graph, is 4-connected but not
5-connected. In this paper, we show that if a Haj\'{o}s graph admits a 4-cut or
5-cut with a planar side then the planar side must be small or contains a
special wheel. This is a step in our effort to reduce Haj\'{o}s' conjecture to
the Four Color Theorem.

20
The categories of this paper are: acc-phys
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
4
The categories of this paper are: cond-mat
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
6
The categories of this paper are: astro-ph
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
3
The categories of this paper are: physics
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The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
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The categories of this paper are: stat
5
The categories of this paper are: gr-qc