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There are $112$ non-isomorphic $6$-vertex planar connected graphs, $81$ of which are $3$-colorable.

I'm searching for one example of an ($n\geq 6$-vertex planar connected graph:

a) that does not contain an even-vertex wheel graph: (W4, W6, W8, W10, etc.)

b) whose vertices are not $3$-colorable

I know that there are plenty of examples, but I can't come up with any.

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    How do you know there are plenty of examples? – 2017-01-21
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    related: https://arxiv.org/pdf/1309.7120.pdf – 2017-01-21
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    @HenningMakholm The paper provided [here](https://pdfs.semanticscholar.org/78a2/bbadb5f83b25a2f91e05f5237ca7e6c65c61.pdf) by Malyshev. – 2017-01-21
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    when you say it contains an even-vertex wheel graph do you mean as an induced subgraph or just as a subgraph? – 2017-01-21
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    @JorgeFernándezHidalgo I should have clarified: as an induced subgraph. – 2017-01-21
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    although now that I think of it, if a planar graph contains a wheel as a subgraph then it also contains a wheel as an induced subgraph. – 2017-01-21
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    If I'm not mistaken there's 99 planar connected 6-vertex graphs. The number 112 that you've given incudes non-planar graphs. Also a quick glance at the paper you cited suggests that it does not assume planarity. – 2017-01-22

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Here's a generated image of all 99 planar connected 6-vertex graphs:

The non-colorable ones have been painted with red vertices. None of them satisfy condition a).

all 99 planar connected 6-vertex graphs

And, there's an image of all 112 (possibly non-planar) connected 6-vertex graphs (note that the enumeration does not match):

Even here, I can't find any graph that satisfies both a) and b).

all 112 connected 6-vertex graphs

So, for $n=6$, there's no such example. For $n=7$, I found several, including these beauties:

                       o-----------o
      o               / \         / \
     /|\             /   \       /   \
    / | \           /     o     o     \
   o--o--o         /   .-  \   /  -.   \
   |\   /|        /. -      \ /      - .\
   | \ / |       o-----------o-----------o
   |  o  |
   | / \ |
   |/   \|
   o-----o
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    I tried different visualization algorithms to make it easier to spot the wheels. Here are the results. Planar: [1](https://i.stack.imgur.com/WVqJG.png) [2](https://i.stack.imgur.com/H3WgV.png) [3](https://i.stack.imgur.com/hLSh6.png) All: [1](https://i.stack.imgur.com/bAxFn.png) [2](https://i.stack.imgur.com/M0zAP.png) [3](https://i.stack.imgur.com/ckLrF.png) – 2017-01-22
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    Is brute force the only viable strategy in order to find a wheel (or 4-clique) in a given graph? – 2017-01-23
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    @EmmaKrentz I'm not an expert in graph theory, so I don't know. – 2017-01-23
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    thank you so much for your help! – 2017-01-23