Hallo,

The task is as follows,

Create a Boolean formula that can be fulfilled if and only if the above graph can be colored with three colors so that adjacent nodes do not have the same color.

Use the variables ai, bi, ci for i ∈ {1,2,3,4,5} to represent the color of a node. The variables ai or bi or ci apply precisely when the node i is colored with color a or b or c.

Exactly one color must be assigned to each node.

Incidentally, I thought a total of 6 options for 3 colors colored in such graph.

*IF I only pay attention to the node, I get 3 implications for each node, then a total of 15 implications.*

(a1-> !a2& !a3)& (a2-> !a1& !a3&!a4)&(a3->!a1& !a2& !a4& !a5)&(a4->!a2& !a3& !a5)&(a5->!a3& !a4)&(b1-> !b2& !b3)& (b2-> !b1& !b3&!b4)&(b3->!b1& !b2& !b4& !b5)&(b4->!b2& !b3& !b5)&(b5->!b3& !b4)&(c1-> !c2& !c3)& (c2-> !c1& !c3&!c4)&(c3->!c1& !c2& !c4& !c5)&(c4->!c2& !c3& !c5)&(c5->!c3& !c4)

it is clear that the following formula is wrong, and connected with OR also wrong,

I tried to formulate a formula out of 6 possibilities.

*((a1->!a2& !a3)&(b2-> !b1& !b3&!b4)&(c3->!c1& !c2& !c4& !c5)&(a4->!a2& !a3& !a5)&(b5->!b3& !b4))|*

*((a1-> !a2& !a3)& (c2-> !c1& !c3&!c4)&(b3->!b1& !b2& !b4& !b5)&(a4->!a2& !a3& !a5)&(c5->!c3& !c4))|*

*((b1->!b2& !b3)&(a2-> !a1& !a3&!a4)&(c3->!c1& !c2& !c4& !c5)&(b4->!b2& !b3& !b5)&(a5->!a3& !a4))|*

*((b1-> !b2& !b3)& (c2-> !c1& !c3&!c4)&(a3->!a1& !a2& !a4& !a5)&(b4->!b2& !b3& !b5)&(c5->!c3& !c4))|*

*((c1-> !c2& !c3)&(a2-> !a1& !a3&!a4)&(b3->!b1& !b2& !b4& !b5)&(c4->!c2& !c3& !c5)&(a5->!a3& !a4))|*

*((c1-> !c2& !c3)&(b2-> !b1& !b3&!b4)&(a3->!a1& !a2& !a4& !a5)&(c4->!c2& !c3& !c5)&(b5->!b3& !b4))*

**everyone is wrong, where should I continue to work？**

**How should I think of this task?**

Thanks for your help in regard.