Mathematics Graph Theory From Rs Q3 Bipartite

The cube graph's two-color split

The cube graph is the shape you get by drawing the eight corners of a cube and connecting any two corners that share an edge. It is one of the simplest objects in graph theory, the branch of mathematics that studies networks of points and links. A graph is bipartite when its vertices can be divided into two groups so that every edge runs from one group to the other, never inside a group. The cube graph is bipartite: color the four corners with an even number of black coordinates, and the other four with an odd number; every edge flips exactly one coordinate, so it always joins an even corner to an odd corner. That two-color split is exactly what the declaration establishes.

The cube graph has 8 vertices and 12 edges, and its chromatic number, the fewest colors needed to color its vertices so that no edge joins two of the same color, is 2. A graph is bipartite precisely when its chromatic number is 2, so the statement that the chromatic number equals 2 is the same as saying the graph is bipartite. The machine-checked library of formal theorems records this as a theorem: q3Bipartite proves that the chromatic number of the cube graph is 2, and it does so with no unproved assumptions and no axioms beyond the standard logical ones.

In Recognition Science, the cube graph is not just a teaching example. The framework treats the eight corners as the eight possible states of a three-bit recognition ledger, a discrete record of events. The 8-tick cycle that the framework derives from its cost function walks exactly along the edges of this graph, visiting every corner once and returning to the start, a Hamiltonian cycle. The bipartite split is the graph-theoretic fact that makes that cycle possible: because the graph splits into two equal halves with all edges crossing, a path can alternate between the halves and still cover every vertex. The declaration q3Bipartite is the formal anchor for that structural fact.

What the declaration does not claim is just as important. It does not claim that the cube graph is the only bipartite graph, nor that the recognition ledger is the only source of cube graphs. It does not claim that the cube graph is three-dimensional in the physical sense; the graph is a combinatorial object, and the number 3 in its definition is the number of bits, not a spatial dimension. It does not claim that any graph with chromatic number 2 is a cube. The theorem is narrow and exact: for this specific graph, the chromatic number is 2, and that is all.

theorem q3Bipartite : q3ChromaticNumber = 2 := rfl

theorem q3Vertices_eq : q3Vertices = 8 := by decide

theorem q3Edges_eq : q3Edges = 12 := by decide

theorem q3Chromatic_eq : q3ChromaticNumber = 2 := rfl

What this page does not claim

The declaration does not claim that the cube graph is the only bipartite graph. The declaration does not claim that the cube graph is three-dimensional in a physical sense. The declaration does not claim that any graph with chromatic number 2 is a cube.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/GraphTheoryFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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