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The \(a\)-graph coloring problem. (English) Zbl 1358.05119
Summary: No proof of the 4-color conjecture reveals why it is true; the goal has not been to go beyond proving the conjecture. The standard approach involves constructing an unavoidable finite set of reducible configurations to demonstrate that a minimal counterexample cannot exist. We study the 4-color problem from a different perspective. Instead of planar triangulations, we consider near-triangulations of the plane with a face of size 4; we call any such graph an \(a\)-graph. We state an \(a\)-graph coloring problem equivalent to the 4-color problem and then derive a coloring condition that a minimal \(a\)-graph counterexample must satisfy, expressing it in terms of equivalence classes under Kempe exchanges. Through a systematic search, we discover a family of \(a\)-graphs that satisfy the coloring condition, the fundamental member of which has order 12 and includes the Birkhoff diamond as a subgraph. Higher-order members include a string of Birkhoff diamonds. However, no member has an applicable parent triangulation that is internally 6-connected, a requirement for a minimal counterexample. Our research suggests strongly that the coloring and connectivity conditions for a minimal counterexample are incompatible; infinitely many \(a\)-graphs meet one condition or the other, but we find none that meets both.

MSC:
05C15 Coloring of graphs and hypergraphs
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