Planar Game Chromatic Numbers and Builder Strategies

Definitions and mathematical statement

For a finite simple graph $G$, the usual vertex-coloring game has Alice and Bob alternate properly coloring one uncolored vertex from a fixed palette. Alice seeks a complete coloring and Bob seeks to prevent it. With the usual Alice-first, no-pass convention, let $\chi_g(G)$ be the least palette size for which Alice can force completion.

The first question is to determine \[ M_{\mathrm{pl}}=\sup\{\chi_g(G):G\text{ is a finite planar graph}\}. \] The source reports $11\leq M_{\mathrm{pl}}\leq17$ and asks for the exact extremal value. This is a bound on the maximum over the class, not a claim that every planar graph has game chromatic number at least $11$. The cited numerical range is retained as supplied historical information, not as an independently updated best bound.

The second question uses a separate online edge-coloring game. Start with an empty graph on an available supply of vertices. At each move Builder chooses a new edge, subject to the requirement that the graph of all chosen edges remains planar; Painter immediately and irrevocably colors that edge red or blue. For a fixed finite graph $H$, Builder seeks to force a monochromatic subgraph isomorphic to $H$ after finitely many moves, regardless of Painter's choices. The copy need not be induced. The proposed characterization is \[ \text{Builder has a winning strategy}\quad\Longleftrightarrow\quad H\text{ is outerplanar}, \] where outerplanar means admitting a plane embedding with every vertex on the boundary of the unbounded face.

The source does not specify the first game's move-order/pass conventions, nor whether the second game's planarity constraint refers only to the current abstract graph or to a fixed embedding. The latter is normally read as abstract planarity, allowing new vertices and without prescribing a single drawing throughout. These conventions are explicit contextual interpretations, not additional verified source conditions.

Short English statement

Determine the largest game chromatic number among planar graphs. Separately, decide whether Builder can force a red or blue copy of a fixed graph while always keeping the board planar exactly for outerplanar targets.

Sources

• Supplied report: Mini-Workshop: Positional Games (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/20, pp. 1073--1114; proposed year 2007. https://doi.org/10.4171/owr/2007/20.
• Local UnsolvedMath registry, permanent ID 30000746, catalogue entry; created 25 August 2026, updated 16 September 2026. The corrected_verified statement asks for the planar maximum, resolving the original fragment's misleading per-graph wording of the $11$--$17$ range. The supplied 21 August 2026 literature triage labels both the exact-value and outerplanarity-characterization questions open. No additional external reference is supplied; neither the bounds nor current status have been independently checked here.