Triangle-overlap cores and density bounds for Tuza's conjecture
Year: 2026
Type: preprint
Abstract: This record contains the paper Triangle-overlap cores and density bounds for Tuza's conjecture by Yasuhito Tsujimura and its finite-verification package. For finite simple graphs, the paper proves Tuza's triangle packing-covering inequality under explicit triangle-overlap and maximum-average-degree conditions. Each edge may belong to at most four triangles; the four-triangle edges form components with at most two edges; and maximum average degree at most 25/2 suffices. No maximum vertex-degree bound is imposed. The paper localizes the density condition to the induced subgraph on the Hifumi core and establishes a quantitative strengthening to the strict bound 25/2 + 2/56775. A separate reduction theorem for a broader overlap class gives a density-free sufficient condition when this core is empty. The general Tuza conjecture remains unresolved. The accompanying verification package contains an offline English reader guide, source code, fixed inputs, finite certificates, and commands for replaying the eight finite proof obligations F001-F008. Start by extracting Tuza_Verification_v1.zip and opening START_HERE.html inside tuza-verification. Full replay has been tested on macOS ARM64 with Apple Clang and Ubuntu ARM64 with GNU GCC. The guide states the measured runtimes, resource requirements, exact checked domains, and the boundary between finite computation and the analytic arguments in the paper. The corresponding paper is publicly available as arXiv:2611.11128v1: https://arxiv.org/abs/2611.11128v1 The paper PDF preserves the submitted version. The verification package is released with verification-use terms in RIGHTS.txt; third-party components retain their respective licenses.
Author: Tsujimura, Yasuhito
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225746982
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