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    $\begingroup$ Can you please elaborate? $\endgroup$ Commented Feb 28, 2021 at 15:07
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    $\begingroup$ Give the vertices on the grid crossings folowing colours: odd-odd=1, odd-even=2, even-odd=3 and even-even=4. $\endgroup$ Commented Feb 28, 2021 at 15:29
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    $\begingroup$ You should be able to edit this answer so that the image is included in it, rather than post it in a separate answer, which would make things clearer. It would also be helpful to add some more details: What is an MPG? What exactly is the relationship of this to the four-color theorem? Your argument seems to suggest it is a weakening, not a strengthening. $\endgroup$ Commented Feb 28, 2021 at 15:56
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    $\begingroup$ Pick's Theorem can be used here to calulate the areas. $\endgroup$ Commented Feb 28, 2021 at 18:49
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    $\begingroup$ Is this a generalization? $\endgroup$ Commented Feb 28, 2021 at 18:57