GPT 6 Astra Ultra attempt on Math-5460

Problem 149: a degeneracy threshold beyond the supplied subclass

1) OBJECTIVE AND SOURCE REVIEW

Attempt dated 2026-09-23. The strong chromatic index conjecture asks for at most $5\Delta^2/4$ colors for even $\Delta$, and $(5\Delta^2-2\Delta+1)/4$ for odd $\Delta$. I read gpt_pro_5.4/149.tex fully. It applies Yu's theorem to 2-degenerate graphs. I retain the general degeneracy parameter to obtain a wider certified region.

2) WORK

The checked theorem of Yu states that a $k$-degenerate graph has strong chromatic index at most \[ U(k,\Delta)=(4k-2)\Delta-k(2k-1)+1. \] This is a published input. Let $k\ge1$ and suppose $k\le5\Delta/16$. Then \[ U(k,\Delta)\le\frac54\Delta^2-2\Delta-2k^2+k+1 \le\frac54\Delta^2-2\Delta. \] The last step uses $-2k^2+k+1\le0$. This is below both parity-dependent conjectured bounds for $\Delta\ge1$. Thus every graph with degeneracy at most $5\Delta/16$ satisfies the conjecture. In particular, a counterexample must have degeneracy strictly greater than $5\Delta/16$, in addition to the restrictions in the supplied attempt.

Keeping the constant term gives a better small example: for $k=3$, $U=10\Delta-14$. For even $\Delta\ge8$, the difference from the target, multiplied by 4, is $5\Delta^2-40\Delta+56>0$; it is 56 at 8 and increasing. For odd $\Delta\ge7$ the corresponding difference is $5\Delta^2-42\Delta+57>0$; it is 8 at 7 and increasing. Therefore all 3-degenerate graphs with $\Delta\ge7$ are also certified, even where the simpler ratio threshold does not apply.

3) ADVERSARIAL VERIFICATION

The odd target is smaller than $5\Delta^2/4$, so it must be checked separately; the displayed inequalities do so. Degree zero is immediate, with no edges to color. Degeneracy three alone does not settle the intermediate maximum degrees 4, 5, and 6 through this estimate: substituting the bound does not justify those cases. These are corollaries of an existing theorem, not new universal coloring algorithms.

4) FINAL

UNRESOLVED. The unsolved regime still includes graphs with degeneracy comparable to their maximum degree, including the extremal constructions. No bound for that whole regime is proved here.

5) SOURCES CHECKED

Complete supplied file; G. Yu, Strong edge-colorings for $k$-degenerate graphs, https://arxiv.org/abs/1212.6093, checked 2026-09-23 for the exact formula.

6) COMPLETION ESTIMATE

Subjective progress toward the universal conjecture. COMPLETION: 15%