Excluding every two-color assignment on a triangle

Goal and scope

Prove chi=3 using an upper witness and complete lower-color search.

Exact enumeration at n=3, with no greedy-optimality assumption.

Givens and symbols

• work

• G1 — V={0,1,2}, E={01,02,12}; two-color labels are 0,1, with assignment vectors ordered by vertices(0,1,2).

• V,E — complete vertex and edge sets
• χ(G) — minimum number of colors

Rules and sources

• mko_definition · cited — One proper k-coloring proves chi(G)≤k; equality also needs a valid exclusion of (k−1)-colorings. Complete small graphs are checked exhaustively for a lower bound and given an explicit coloring for an upper bound. A greedy result alone is not optimality evidence. The empty vertex graph has chi=0 and a nonempty edgeless graph has chi=1. · mko-chromatic-number
• work · prerequisite — A finite simple undirected graph has loop-free duplicate-free unordered edges; graph properties are audited on all declared V and E. A proper coloring gives distinct colors across every edge; chi is the minimum used-color count over all total proper colorings, and k-colorable means using at most k colors.

Main derivation

c=(0,1,2)

c(0)\ne c(1),\quad c(0)\ne c(2),\quad c(1)\ne c(2)

c:V\to\{0,1\}

2^3=8\text{ assignments}

000,001

c(0)=c(1)=0\implies01\text{ violated}

010,101

c(0)=c(2)\implies02\text{ violated}

011,100

c(1)=c(2)\implies12\text{ violated}

110,111

c(0)=c(1)=1\implies01\text{ violated}

\text{no proper map }V\to\{0,1\}

\chi(G)>2

2<\chi(G)\le3,\quad\chi(G)\in\mathbb N

\chi(G)=3

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — The separate conventions are chi=0 for no vertices and chi=1 for nonempty edgeless graphs.
• S07 · premise_audit · G1, S06, work — This is a global lower bound, not one failed candidate.
• S06 · premise_audit · G1, S05, work — The set is 000,001,010,011,100,101,110,111.
• S05 · premise_audit · G1, S04, work — Swapping color labels does not remove the violation.
• S04 · premise_audit · G1, S03, work — Check02 too; it is an edge despite nonconsecutive labels.
• S03 · premise_audit · G1, S02, work — Both cases are explicitly listed.
• S02 · premise_audit · G1, S01, work — At most two colors does not require both to appear.
• S01 · premise_audit · G1, work — Equality still requires excluding fewer colors.

Error controls and negative controls

Complete enumeration

Each of the eight bit assignments appears once.

Bounds

Give three colors, exclude every two-color assignment, then use integrality.

• A three-coloring alone proves chi=3. → reject — It gives only an upper bound; the eight-case exclusion supplies the lower bound.
• 010 is proper on the triangle. → reject — Edge 02 has color0 at both ends.
• A nonempty edgeless graph has chi=0. → reject — Each vertex still needs a color, so the minimum is 1.

Result and limitations

All eight two-color cases fail and a three-color witness exists, hence chi=3.

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: e067a4fb1baa6f694ba27da6d3b5d80c12c74434d72f796ad27dd14c604b48ad
• original source SHA-256: 1ca25d8841495fe4e298468a1912e58c28402ed9745e1fee11300a1b7c39f7b5
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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