A graph coloring game
This is a 2-player game played by players $A$ and $B$. A random non-trivial planar connected graph $G(V,E)$ is chosen. Player $A$ sets up the game as follows:
Player $A$ partitions the graph into a spanning tree $T$ of his choice and the set of edges $E_\Delta$ that contains the edges removed from $G$ to create $T$. The edges in $E_\Delta$ are coalesced into connected components. Let the set of connected components be $C_\Delta$. Observe that $c \in C_{\Delta}$ may have cycles (eg: $K_4$ with $C_3$ extracted from it). In that case Player $B$ may optionally split the cycles so that $E_{\Delta}$ has cycle-free components. This is done at the start of the game only and may not be done after the game is in progress.
Player $A$ chooses two colors, say $\{red, blue\}$ and Player B chooses two additional colors, i.e., $\{red, blue, green, yellow\}$.
Player $A$ colors $T$ with his chosen colors in any way as long as adjacent vertices don't have the same color.
Player $B$ colors each $c \in C_{\Delta}$ with his chosen colors so that adjacent vertices in $c$ don't have the same color.
The game is completely played by player $B$ from this point onwards.
Player $B$ now chooses a $c \in C_{\Delta}$ and merges it with $T$ (From this point we will refer to the merged subgraph as $T$ although it may not be a tree anymore). During a merge, Player $B$ can only change the colors in the subgraph $c$ that he is merging (and not the color of any vertex in $T$ other than the common vertices $V(T) \cup V(c)$). In any stage in the game if a merge results in conflicts for any other $c^\prime \in C_{\Delta}, c^\prime \ne c$, Player $B$ must resolve the conflict by recoloring $c^\prime$. If this recoloring of $c^\prime$ requires re-coloring $T$, player $B$ is allowed to re-color $T$. Note that this is the only condition under which player $B$ is allowed to re-color $T$.
Player $B$ wins if $G$ is colored with $4$ or fewer colors.
Player $B$ loses (and $A$ wins) if at any stage, a conflict cannot be resolved.
Player $A$ wins by default if Player $B$ forfeits.
If one of the players has a winning strategy, what is it?
This question was inspired by the Four-Color theorem and the elusiveness of a simple proof for it. Although this is not the main question, I was wondering if one could prove the Four-Color theorem using a series of merges of the spanning tree $T$ with $c \in C_{\Delta}$ and 4-color recolorings of intermediate merged subgraphs and if there was a way to do it systematically without triggering infinite re-coloring loops.