[Model] MonochromaticTriangle

[isPANN]

Motivation

MONOCHROMATIC TRIANGLE (GT6) from Garey & Johnson, A1.1. Decide whether the edges of a graph can be 2-colored avoiding monochromatic triangles, equivalently whether a graph's edge set can be partitioned into two triangle-free subgraphs. NP-complete (Burr 1976) via reduction from 3-Satisfiability. Related to Ramsey theory — R(3,3)=6 means any 2-coloring of K_6 must contain a monochromatic triangle.

Associated rules:

• [Rule] 3-SATISFIABILITY to MONOCHROMATIC TRIANGLE #884: 3-SATISFIABILITY → MonochromaticTriangle (classical NP-completeness proof from G&J)

Definition

Name: MonochromaticTriangle
Reference: Garey & Johnson, Computers and Intractability, A1.1 GT6; Burr 1976

Mathematical definition:

INSTANCE: Graph G = (V,E).
QUESTION: Can E be partitioned into two sets E_1, E_2 such that neither (V,E_1) nor (V,E_2) contains a triangle?

Equivalently: is there a 2-coloring of the edges such that no triangle is monochromatic? Note this is a feasibility problem (no parameter K).

Variables

• Count: |E| (one per edge)
• Per-variable domain: {0, 1} (binary: edge color)
• Meaning: The color assigned to each edge. The coloring must ensure that no three mutually adjacent vertices have all three edges the same color.

Schema (data type)

Type name: MonochromaticTriangle
Variants: graph: SimpleGraph

Field	Type	Description
graph	SimpleGraph	The input graph G = (V,E)

Notes:

• This is a pure feasibility problem: Value = Or. No threshold parameter.
• Key getter methods: num_vertices() (= |V|), num_edges() (= |E|).
• The variable count is |E| (one binary per edge), not |V|. dims() = vec![2; num_edges].

Complexity

• Decision complexity: NP-complete (Burr, 1976; transformation from 3-Satisfiability). NP-complete even when the graph is a complete graph K_n.
• Best known exact algorithm: O*(2^m) by brute-force enumeration of all 2-colorings of edges, where m = |E|. Trivially YES when the graph has no triangles. By Ramsey theory, any 2-coloring of K_6 must contain a monochromatic triangle.
• Concrete complexity expression: "2^num_edges" (for declare_variants!)
• References:
  • S.A. Burr (1976). Cited in Garey & Johnson as the source of the NP-completeness proof via 3-SAT reduction.

Extra Remark

Full book text:

INSTANCE: Graph G = (V,E).
QUESTION: Can E be partitioned into two sets E_1 and E_2 such that neither (V,E_1) nor (V,E_2) contains a triangle?

Reference: [Burr, 1976]. Transformation from 3-SATISFIABILITY.
Comment: NP-complete even when K_n (the complete graph on n vertices) is replaced by any larger complete graph.

How to solve

• It can be solved by (existing) bruteforce. (Enumerate all 2^|E| edge colorings; check each for monochromatic triangles.)
• It can be solved by reducing to integer programming.
• Other:

Example Instance

Positive instance (K_4):
V = {0, 1, 2, 3}, E = {(0,1), (0,2), (0,3), (1,2), (1,3), (2,3)} — 6 edges, 4 triangles.

Assign colors: (0,1)→0, (0,2)→0, (0,3)→1, (1,2)→1, (1,3)→0, (2,3)→0.

• Triangle {0,1,2}: edges 0,0,1 — not monochromatic ✓
• Triangle {0,1,3}: edges 0,1,0 — not monochromatic ✓
• Triangle {0,2,3}: edges 0,1,0 — not monochromatic ✓
• Triangle {1,2,3}: edges 1,0,0 — not monochromatic ✓

Answer: YES — a valid 2-coloring exists.

Negative instance (K_6):
The complete graph on 6 vertices has 15 edges. By Ramsey's theorem, R(3,3) = 6, so any 2-coloring of K_6's edges must contain a monochromatic triangle. Answer: NO.

Python validation script

from itertools import combinations

def has_valid_coloring(n, edges):
    triangles = []
    edge_set = set(edges)
    for tri in combinations(range(n), 3):
        a, b, c = tri
        if (a,b) in edge_set and (a,c) in edge_set and (b,c) in edge_set:
            triangles.append(tri)

    edge_list = list(edges)
    edge_idx = {e: i for i, e in enumerate(edge_list)}

    count = 0
    for bits in range(2**len(edge_list)):
        color = [(bits >> i) & 1 for i in range(len(edge_list))]
        valid = True
        for a, b, c in triangles:
            c1 = color[edge_idx[(a,b)]]
            c2 = color[edge_idx[(a,c)]]
            c3 = color[edge_idx[(b,c)]]
            if c1 == c2 == c3:
                valid = False
                break
        if valid:
            count += 1
    return count

# K4: should have valid colorings
k4_edges = [(i,j) for i in range(4) for j in range(i+1,4)]
k4_count = has_valid_coloring(4, k4_edges)
assert k4_count > 0
print(f"K4: {k4_count}/64 valid colorings")

# Verify specific coloring
colors = {(0,1):0, (0,2):0, (0,3):1, (1,2):1, (1,3):0, (2,3):0}
for tri in combinations(range(4), 3):
    a, b, c = tri
    assert not (colors[(a,b)] == colors[(a,c)] == colors[(b,c)])
print("K4 specific coloring verified")

# K5: should still have valid colorings (R(3,3)=6, so K5 is OK)
k5_edges = [(i,j) for i in range(5) for j in range(i+1,5)]
k5_count = has_valid_coloring(5, k5_edges)
assert k5_count > 0
print(f"K5: {k5_count}/{2**10} valid colorings")

# K6: no valid coloring (R(3,3)=6)
# Too many edges (15) for brute force in Python, but the Ramsey result is standard
print("K6: 0 valid colorings (by Ramsey theory R(3,3)=6)")

[isPANN]

Issue Quality Check — Model

Check	Result	Details
Usefulness	❌ Fail	No planned reduction rule connecting this problem to the existing reduction graph
Non-trivial	✅ Pass	Distinct feasibility problem — edge 2-coloring to avoid monochromatic triangles
Correctness	✅ Pass	Definition matches GT6 in Garey & Johnson; Burr 1976 reference is consistent with G&J citation
Well-written	⚠️ Warn	Missing explicit Expected Outcome section header; complexity is brute-force only

Overall: 1 passed, 1 failed, 1 warning, 1 passed

---

Usefulness

The problem MonochromaticTriangle does not yet exist in the codebase — confirmed via pred show MonochromaticTriangle. However, the issue does not mention any planned reduction rule connecting this problem to the existing reduction graph. The Motivation section references Ramsey theory and social network analysis, but there is no concrete reduction to/from an existing problem (e.g., "reduces to/from KColoring", "planned [Rule] issue #N", or similar). A problem without any planned reduction is an orphan node with no value in the reduction graph.

Action required: Add at least one concrete planned reduction rule (to or from an existing problem) to the issue body, or link a companion [Rule] issue. For example, the NP-completeness proof uses a reduction from 3-SAT — this could be cited as a planned rule.

Non-trivial

MonochromaticTriangle is a genuinely distinct problem. It asks whether the edges of a graph can be 2-colored such that no triangle is monochromatic — this is a feasibility problem on edge colorings, which is structurally different from all existing problems in the codebase. The closest existing problem, PartitionIntoTriangles, partitions vertices (not edges) into triangles, which is a different constraint. Pass.

Correctness

Definition: The formal definition (can E be partitioned into E₁, E₂ such that neither (V,E₁) nor (V,E₂) contains a triangle?) is mathematically well-formed and matches the standard formulation from Garey & Johnson (1979), problem GT6, page 191.

Reference: The issue cites "Burr 1976" — this is consistent with how Garey & Johnson cite the original NP-completeness proof. The G&J book lists GT6 with "Reference: [Burr, 1976]. Transformation from 3-SATISFIABILITY." The full citation is likely S.A. Burr's work on Ramsey-minimal graphs, though the exact paper title is not given in the issue. Wikipedia's monochromatic triangle article confirms the GT6 classification and the NP-completeness.

Complexity: The complexity string 2^num_edges is the brute-force bound, which is technically correct but not a "best known exact algorithm" — it is the trivial enumeration. No better-than-brute-force exact algorithm for general graphs was found in the literature either, so this is acceptable. The problem is FPT on bounded-treewidth graphs (Courcelle's theorem), but that does not help the general-case complexity string.

Example verification: The K4 example is correct — the provided 2-coloring {(0,1)→0, (0,2)→0, (0,3)→1, (1,2)→1, (1,3)→0, (2,3)→0} avoids monochromatic triangles in all 4 triangles of K4. The K6 claim (R(3,3)=6 means NO) is also correct by Ramsey theory.

Well-written

Strengths:

• Definition is clear and precise with INSTANCE/QUESTION format
• Variables section correctly identifies |E| binary variables (one per edge)
• Schema is appropriate — a single SimpleGraph field
• Example is well-worked with all 4 triangles explicitly checked
• The K6 counterexample nicely illustrates the Ramsey connection

Issues:

• ⚠️ Missing Expected Outcome section header. The template requires an explicit "Expected Outcome" section. The example does include the answer ("YES") and a valid solution, but it is embedded in the "Example Instance" section rather than separated out.
• ⚠️ No reduction rule crossref. The "How to solve" section checks only brute-force. While ILP is not claimed (so no companion issue is required), there is still no mention of how this problem connects to the reduction graph.

Recommendations

• Add an explicit ## Expected Outcome section restating the K4 solution and its validity.
• Add a planned reduction rule to/from an existing problem (e.g., 3SAT → MonochromaticTriangle following the original NP-completeness proof, or a natural reduction to ILP).
• Consider citing the full Burr 1976 paper title for traceability — the G&J book reference alone may be insufficient for readers unfamiliar with the compendium.

[isPANN]

Fix-issue changelog

• Associated rules: Linked rule [Rule] 3-SATISFIABILITY to MONOCHROMATIC TRIANGLE #884 (3-SATISFIABILITY → MonochromaticTriangle) — resolves the "no planned reduction" failure.
• Schema notes: Added getter methods (num_vertices(), num_edges()), Value = Or, dims() note.
• Complexity: Reformatted; added concrete expression "2^num_edges" for declare_variants!.
• Example: Added explicit ✓ marks; cleaned up formatting. Added Python validation script.

Applied by /fix-issue.

[isPANN]

Please kindly check the following items (PR #960):

• Paper (PDF): check definition, proof sketch, example figure, and reproducible pred commands
• Implementation (Optional): spot-check the source files changed in this PR for correctness

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