Graph Coloring Game Calculator

χ_g(G) game chromatic

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About Graph Coloring Game Calculator

A graph coloring game calculator computing χ_g(G): minimum colors for Alice to guarantee proper coloring against adversarial Bob. χ(G) ≤ χ_g(G) ≤ 2χ(G). Forests: χ_g ≤ 4. Planar: χ_g ≤ 17. Game-theoretic graph coloring. Client-side.

Graph Coloring Game Calculator Features

  • χ_g(G) value
  • Alice vs Bob
  • Tree bound
  • Planar bound
  • Game simulation
Graph coloring game: Alice and Bob alternately color vertices properly. Alice wants to complete; Bob wants to force a stuck position. χ_g(G) = min colors for Alice to win. χ(G) ≤ χ_g(G). Bodlaender (1991) initiated the study.

How to Use

Select graph:

  • χ_g: Game chromatic #
  • Strategy: Alice's plan
  • Bounds: χ ≤ χ_g ≤ 2χ

Known Bounds

Forests: χ_g ≤ 4 (Faigle et al., 1993). Outerplanar: χ_g ≤ 7. Planar: χ_g ≤ 17 (Zhu, 2008). Partial k-trees (tw≤k): χ_g ≤ 3k+2. Complete graphs: χ_g = χ = n (trivially).

Game Variants

Marking game: competitive coloring where colors aren't specified. Activation game. Maker-Breaker coloring game. Each variant captures different adversarial aspects. Rich game-theoretic structures.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2Determine χ_g.
  3. 3Find Alice strategy.
  4. 4Test with Bob.
  5. 5Compare χ vs χ_g.

Graph Coloring Game Calculator — Frequently Asked Questions

Why is χ_g > χ sometimes?+

Bob can force Alice into bad positions! Even on a forest (χ=2), Bob can force 4 colors needed. Bob doesn't care about completing the coloring — he strategically creates conflicts that require more colors.

Is the game chromatic number computable?+

PSPACE-complete in general (like most 2-player games). For specific graph families: known bounds. For small graphs: exhaustive game tree analysis. No polynomial algorithm is known or expected.

What's the gap between χ and χ_g?+

Can be large! χ_g(G) ≤ 2χ(G) (greedy bound). For forests: χ=2 but χ_g can be 4. For bipartite graphs: χ=2 but χ_g can be arbitrarily large! The gap measures 'adversarial difficulty'.

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