Excluding bichromatic vertices

Theorem 1 A 4-coloring of the unit plane cannot contain a bichromatic vertex (a vertex that can be colored in either of two colors while keeping the color of all other vertices fixed).

Proof Without loss of generality we may assume that the colors are [math]\displaystyle{ {\bf Z}/4{\bf Z} }[/math] and that the origin is bichromatic with colors 0,2.

Let [math]\displaystyle{ \omega := \exp( i \pi/3) }[/math] be the sixth root of vector, then we have the identities

[math]\displaystyle{ \omega^6 = 1; \quad \omega^3 = -1; \quad \omega^2 = \omega - 1. \quad (1) }[/math]

Let [math]\displaystyle{ \eta := \exp( \frac{i}{2} \arccos(\frac{5}{6}), }[/math] then [math]\displaystyle{ \eta }[/math] is a unit vector that obeys the identity

[math]\displaystyle{ \eta (\omega + \omega^2) - \overline{\eta} (\omega + \omega^2)+ 1 = 0 \quad (2). }[/math]

Let

[math]\displaystyle{ C_6 := \{ \omega^j: j=0,\dots,5\} }[/math]

denote the multiplicative cyclic group of order 6. We consider the coloring of the following seven cosets of [math]\displaystyle{ C_6 }[/math]:

[math]\displaystyle{ C_6, (\eta - \overline{\eta}) C_6, (\eta - \overline{\eta}\omega ) C_6, \eta C_6, (1+\eta\omega^2 ) C_6, \overline{\eta} C_6, (1 + \overline{\eta \omega^2 }) C_6. }[/math]

Clearly [math]\displaystyle{ C_6 }[/math] is conjugation invariant. Since

[math]\displaystyle{ \overline{\eta - \overline{\eta}} = (\eta - \overline{\eta}) \omega^3 }[/math]

[math]\displaystyle{ \overline{\eta - \omega \overline{\eta}} = (\eta - \omega \overline{\eta}) \omega^2 }[/math]

we also see that [math]\displaystyle{ (\eta - \overline{\eta}) C_6 }[/math], [math]\displaystyle{ (\eta - \overline{\eta}\omega C_6 }[/math] are conjugation invariant, while [math]\displaystyle{ \eta C_6 }[/math] is conjugate to [math]\displaystyle{ \overline{\eta} C_6 }[/math] and [math]\displaystyle{ (1+\eta \omega^2 ) C_6 }[/math] is conjugate to [math]\displaystyle{ (1 + \overline{\eta \omega^2 }) C_6 }[/math].

We observe the following unit edges:

1. Within [math]\displaystyle{ C_6 }[/math], there is a unit edge from any [math]\displaystyle{ \omega^j \in C_6 }[/math] to the origin, to [math]\displaystyle{ \omega^{j+1} \in C_6 }[/math], and [math]\displaystyle{ \omega^{j-1} \in C_6 }[/math].
2. Within [math]\displaystyle{ \eta C_6 }[/math], there is a unit edge from any [math]\displaystyle{ \eta \omega^j \in C_6 }[/math] to the origin, to [math]\displaystyle{ \eta \omega^{j+1} \in \eta C_6 }[/math], and [math]\displaystyle{ \eta \omega^{j-1} \in \eta C_6 }[/math]. Similarly with [math]\displaystyle{ \eta }[/math] replaced by [math]\displaystyle{ \overline{\eta} }[/math].
3. Using (2) (which can be rewritten for instance as [math]\displaystyle{ (\eta - \overline{\eta}) (\omega^2 - 1) = -\omega }[/math]), we see that within [math]\displaystyle{ (\eta - \overline{\eta}) C_6 }[/math], there is a unit edge from any [math]\displaystyle{ (\eta - \overline{\eta}) \omega^j }[/math] to [math]\displaystyle{ (\eta - \overline{\eta}) \omega^{j+2} }[/math] and [math]\displaystyle{ (\eta - \overline{\eta}) \omega^{j-2} }[/math].
4. Each [math]\displaystyle{ (\eta - \overline{\eta}) \omega^j \in (\eta - \overline{\eta}) C_6 }[/math] has a unit edge to [math]\displaystyle{ (\eta - \overline{\eta} \omega) \omega^j \in (\eta - \overline{\eta} \omega) C_6 }[/math], to [math]\displaystyle{ (\eta - \overline{\eta} \omega) \omega^{j-1} \in (\eta - \overline{\eta} \omega) C_6 }[/math], to [math]\displaystyle{ \eta \omega^j \in \eta C_6 }[/math], to [math]\displaystyle{ -\overline{\eta} \omega^j \in \overline{\eta} C_6 }[/math], to [math]\displaystyle{ \eta \omega^j - \overline{\eta} (\omega^j + \omega^{j+1}) = (1 + \eta \omega^2) \omega^{j+2} \in (1+\eta \omega^2) C_6 }[/math], and to [math]\displaystyle{ \eta (\omega^j + \omega^{j-1}) - \overline{\eta} \omega^j = (1+\overline{\eta \omega^2}) \omega^{j+1} \in (1 + \overline{\eta \omega^2}) C_6 }[/math]. (Here (2) is used to obtain the latter two identities.)
5. Each [math]\displaystyle{ (\eta - \overline{\eta} \omega) \omega^j \in (\eta - \overline{\eta} \omega) C_6 }[/math] has a unit edge to [math]\displaystyle{ \eta \omega^j \in \eta C_6 }[/math], to [math]\displaystyle{ -\overline{\eta} \omega^{j+1} \in \overline{\eta} C_6 }[/math], to [math]\displaystyle{ \eta \omega^j - \overline{\eta} (\omega^j + \omega^{j+1}) = (1 + \eta \omega^2) \omega^{j+2} \in (1+\eta \omega^2) C_6 }[/math], and to [math]\displaystyle{ \eta (\omega^{j+1} + \omega^j) - \overline{\eta} \omega^{j+1} = (1+\overline{\eta \omega^2}) \omega^{j+2} \in (1 + \overline{\eta \omega^2}) C_6 }[/math]. (Again, the latter two identities come from (2).)
6. Each [math]\displaystyle{ \omega^j \in C_6 }[/math] has a unit edge to [math]\displaystyle{ \omega^j + \eta \omega^{j+2} \in (1 + \eta \omega^2) C_6 }[/math] and to [math]\displaystyle{ \omega^j + \overline{\eta} \omega^{j-2} \in (1 + \overline{\eta \omega^2}) C_6. }[/math]
7. Each [math]\displaystyle{ \eta \omega^j \in C_6 }[/math] has a unit edge to [math]\displaystyle{ \omega^{j-2} + \eta \omega^j \in (1 + \eta \omega^2) C_6 }[/math]; similarly, each [math]\displaystyle{ \overline{\eta} \omega^j \in C_6 }[/math] has a unit edge to [math]\displaystyle{ \omega^{j+2} + \overline{\eta} \omega^j \in (1 + \overline{\eta \omega^2}) C_6 }[/math].