Files needed for a verification of the proofs of Lemmas 2.4 and 2.5 in the paper "Degree conditions forcing directed cycles" by Grzesik and Volec

This web-page contains the matrices and the scripts that can be used to verify the correctness of the proofs of Lemmas 2.4 and 2.5 presented in the paper "Degree conditions forcing directed cycles" authored by Andrzej Grzesik and Jan Volec. A preliminary version of the paper can be downloaded from arXiv.

Verification scripts and data (gzipped archive; its compressed size is around 2.5M)

Usage: extract the files from the archive and run sage verify_bounds.sage

Tested with SAGE 9.x (python3-based)

Description of the files in the archive:

• lemma24_files/
  • types_list – the complete list of 34 flag-types s₀, …, s₃₃ used in the proof of Lemma 2.4
  • graph_lists/ – a directory with the list of all the 6-vertex {C₃,C₆}-free oriented graphs as well as the lists of the appropriate s_i-flags
  • cs_0, …, cs_33 – the pairing densities for the flag-types s₀, …, s₃₃
  • M_0, …, M_33 – weighted Sum-of-Squares representation of the PSD matrices used in the proof for the flag-types s₀, …, s₃₃
  • deg_0, …, deg_568 – the coefficients describing the flag-expression [ (10000*out_degree - 2509) × F_j ] = 0
  • w_deg – the 569 weights used in the proof for flag-expressions of the form [ w_j × (10000*out_degree - 2509) × F_j ] = 0
  • verify.sage – the verification script for Lemma 2.4 (in SAGE)
• lemma25_files/
  • types_list – the complete list of 34 flag-types s₀, …, s₃₃ used in the proof of Lemma 2.5
  • graph_lists/ – a directory with the list of all the 6-vertex {C₃,C₄,C₆}-free oriented graphs as well as the lists of the appropriate s_i-flags
  • cs_0, …, cs_33 – the pairing densities for the flag-types s₀, …, s₃₃
  • M_0, …, M_33 – weighted Sum-of-Squares representation of the PSD matrices used in the proof for the flag-types s₀, …, s₃₃
  • deg_0, …, deg_393 – the coefficients describing the flag-expression [ (100000*out_degree - 20009) × F_j ] = 0
  • w_deg – the 394 weights used in the proof for flag-expressions of the form [ w_j × (100000*out_degree - 20009) × F_j ] = 0
  • verify.sage – the verification script for Lemma 2.5 (in SAGE)
• verify_bounds.sage – the main verification script (in SAGE)