What does this research mean for the field?

The split-graph case of Erdős Problem #81 is resolved at the conjectured quadratic scale by proving that the clique-partition number cp(G) ≤ n²/6 + O(n) for every split graph G. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To resolve Erdős Problem #81 regarding the asymptotic edge clique-partition number for chordal graphs, with a specific focus on establishing bounds for split graphs.

August 24, 2026Open Access

Erdős Problem #81 — Chordal Clique Partitions: Papers I–III

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JGJuan Pablo Traverso Gianini

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Implication

Mathematical analysis demonstrates sharp quadratic clique partition bounds in split graphs, resolving a key subcase of Erdős Problem #81.

Key Points

To resolve Erdős Problem #81 regarding the asymptotic edge clique-partition number for chordal graphs, with a specific focus on establishing bounds for split graphs.
Analyzed extremal properties of fractional triangle-packing and fractional triangle-cover functionals across chordal and split graph classes.
Coupled mathematical derivations with interactive theorem formalizations in Lean 4 and verified the proofs through independent automated audit workflows.
Proved that the edge clique-partition number satisfies cp(G) ≤ n²/6 + O(n) for every n-vertex split graph G, establishing the sharp quadratic coefficient of 1/6.
Determined that the exact maximum of |E(G)| − 2τ₃*(G) over n-vertex chordal graphs is ⌊(2n+1)²/24⌋, attained by complete-split graphs, and bounded split graph fractional packings by |E(G)| − 2ν₃*(G) ≤ n²/6 + n.

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Juan Pablo Traverso Gianini (2026) studied this question.

synapsesocial.com/papers/6a8c23d1bca056c88e6dfeaa https://doi.org/10.5281/zenodo.22064657

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Implication

Mathematical analysis demonstrates sharp quadratic clique partition bounds in split graphs, resolving a key subcase of Erdős Problem #81.

Erdős Problem #81 — Chordal Clique Partitions: Papers I–III

Discussion

Member takes

Implication

Key Points

Cite This Study

Erdős Problem #81 — Chordal Clique Partitions: Papers I–III

Discussion

Member takes

Implication

Key Points

Cite This Study

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