## Files needed for a verification of the proofs of Claims A.1 and B.1 in the paper "Minimum number of edges that occur in odd cycles" by Grzesik, Hu, and Volec

This web-page contains the matrices and the scripts that can be used to verify the correctness of the proofs of Claims A.1 and B.1 presented in the
paper Minimum number of edges that occur in odd cycles authored by Andrzej Grzesik, Ping Hu, and Jan Volec.
A preliminary version of the paper can be downloaded from arXiv.

Verification scripts and data (tar; the size of the archive is around 800K)

Usage: extract the files from the archive and run *sage verify_claims.sage*

Description of the files in the archive:

**verify_bounds.sage** – * the main verification script (in SAGE)*
**claimA1_files**
- verify_claim.sage –
*the verification script for Claim A.1 (in SAGE)*
- L_hat, B_hat, R_hat –
*the positive definite matrices L^, B^ R^*
- M_lambda, M_beta, M_rho –
*the transformation matrices M*^{λ}, M^{β} and M^{ρ}
- cs_lambda, cs_beta, cs_rho –
*the pairing densities for the types λ, β and ρ*
- trg_times_rededgebound –
*the H*_{6} coefficients in the expression red-triangle × (8red-edge - 2 - √2)
- indep3_times_halfdensity –
*the H*_{6} coefficients in the expression 3-independent × (red-edge + blue-edge - non-edge)
- cocher_times_halfdensity –
*the H*_{6} coefficients in the expression red-cocherry × (red-edge + blue-edge - non-edge)
- indC4X-coordinates –
*all the coordinates from the list of flags H*_{6} containing C^{X}_{4} as an induced subgraph
- indP5-coordinates –
*all the coordinates from the list of flags H*_{6} containing P_{5} as an induced subgraph
- graph_lists/ –
*directory with the list of all flags in H*_{6}, H^{λ}_{4}, H^{β}_{4} and H^{ρ}_{4}

**claimB1_files**
- verify_claim.sage –
*the verification script for Claim B.1 (in SAGE)*
- L_hat, B_hat, R_hat –
*the positive definite matrices L^, B^, R^*
- M_lambda, M_beta, M_rho –
*the transformation matrices M*^{λ}, M^{β} and M^{ρ}
- cs_lambda, cs_beta, cs_rho –
*the pairing densities for the types λ, β and ρ*
- trg_times_rededgebound –
*the H*_{6} coefficients in the expression red-triangle × (9red-edge - 4)
- indep3_times_halfdensity –
*the H*_{6} coefficients in the expression 3-independent × (red-edge + blue-edge - non-edge)
- cocher_times_halfdensity –
*the H*_{6} coefficients in the expression red-cocherry × (red-edge + blue-edge - non-edge)
- indP4-coordinates –
*all the coordinates from the list of flags H*_{6} containing P_{4} as an induced subgraph
- graph_lists/ –
* directory with the list of all flags in H*_{6}, H^{λ}_{4}, H^{β}_{4} and H^{ρ}_{4}