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If this is in the wrong forum can a mod move it please?

On to business.
Our local gaming group is wrapping up our PE campaign and I've been tasked with modifying the rules for the next campaign. One thing we want is to have everyone's empires be contiguous from the get-go. Normally in PE you pick a start spot, then expand outwards, and when there are no adjacent tiles left to expand into you have to take a spot waaay across the map which is isolated from the rest of your empire. Normally these spots get gobbled up really fast and the people who got map position (usually by going last thanks to dice) such that their empire is seriously fragmented are killed off fast and have less fun. That being the case, for our next go, we're going to try have everyone's empires be contiguous like i mentioned. The logistical problem we've run into with this is that by doing so, with the nature of the hexagonal tiles, you'll really only border on two, maybe three people, if the map building is enforced so that nobody gets cut off.
My question is, from a math-geometry-logic point of view, if there a way to arrange hexagons, colouring them in 6 different colours, so that you border on more than two other teams, while keeping each team's empire intact. Here's a pic of what I mean.



Expanding outwards from that you would end up with something like this if you placed a restriction that nobody can cut off anyone else's expansion.



Which is pretty boring looking.

It also assumes either
a) a roughly isotropic distribution of tiles, or
b) a map setup system in which, after beginning with the original 6 tiles, players took turns adding tiles to the board adjacent to their own, which also belonged to them.

In the regular PE rules, where the map is determined before the players pick tiles, is there a way to ensure that each person can have a equal sized contiguous empire? The only way i could reckon to do this is to have someone act as arbitrator and carve the map up into equal sized portions, and then just have the players roll dice to see what portion then belongs to them. This is a fair way to do it for a cool looking map (not a circle), and allows for possibly more border contact, but removes the personalization and feeling of ownership from the players' empires.

Here is an example of what i mean:

The players set up this map:



and then someone carves it up like this:



The players them roll d6 for what colour they are.

In this way everyone has at least 3 neighbours and a contiguous empire, without the map being a circle/hexagon of outward expanding empires.

Thoughts on these ideas?
And if anyone has another good way to set this up so people's empires aren't fractured, please let me know, cause this is really the last stumbling block in our revision of the rules for our next campaign.