Color identical pairs and the 4-color theorem

Question

If one could prove that for every 4-chromatic planar graph $X$ every color identical pair in $X$ is separated by a cycle, would that be a proof of the 4-color theorem?

Explanation:

A pair of vertices $\{u, v\}$ in a $k$-chromatic graph $G$ is a color identical pair iff the color of $u$ equals the color of $v$ in every $k$-coloring of $G$, and the supergraph of $G$ where $u$ and $v$ are adjacent, $(V(G), E(G) \cup \{uv\})$, then requires $k + 1$ colors.

For every $k$-critical graph $X$ and for every edge $xy$ in $X$, there is a $(k - 1)$-chromatic graph $Y = (V(X), E(X) \setminus \{xy\})$ where $\{x, y\}$ is a color identical pair (because, if there were a $(k - 1)$-coloring of $Y$ where $x$ and $y$ had different colors, that would also have been a $(k - 1)$-coloring of $X$).

E.g. removing an edge from $K_{5}$ gives a 4-chromatic graph of two tetrahedrons whose intersection is a triangle, and the two vertices that are not in the intersection are a color identical pair. On the plane these vertices are separated by the triangle.

Answer 1

The claim appears to be equivalent to the four colour theorem. In the following I'll assume that "$x$ and $y$ can be separated by a cycle" is equivalent to "adding the edge between $x$ and $y$ produces a non-planar graph". (Perhaps this is what your question is actually about: formally showing that a reasonable definition of separating by a cycle captures the intended idea.)

So suppose that $G$ is a planar graph with chromatic number at least 5. Without loss of generality, $G$ is 5-critical. Choose an edge $e=xy$ of $G$ and consider $G-e$. This is 4-chromatic, and in every 4-colouring $x$ and $y$ must receive the same colour (else we have a 4-colouring of $G$). So by the claim, $x$ and $y$ can be separated by a cycle, contradicting the planarity of $G$.

In the other direction, suppose there is a 4-chromatic planar graph $G$ with colour identical vertices $x$ and $y$ such that $x$ and $y$ cannot be separated by a cycle. Then $G + xy$ is a 5-chromatic planar graph.

On being hesitant about claiming anything about the four colour theorem: you're reasonably safe making statements of the form "Claim X is equivalent to the four colour theorem". You only need to start to worry if you think you also have a simple proof of Claim X.