Cycle-residue stability at minimum degree five

The preprint claims an exact classification. Write $C_5(G)$ for the residues modulo five represented by cycle lengths in $G$. Let $\mathcal E_5={K_6,K_{5,5}}\cup{H_{5,n;t}:2\le t\le5<n}$, where $H_{5,n;t}$ is obtained from $K_{5,n}$ by deleting $5-t$ edges incident with one vertex in the part of size $n$. For every finite simple graph $G$ with minimum degree at least five, exactly one alternative holds: $C_5(G)=\mathbb Z_5$; or every end-block belongs to $\mathcal E_5$ and every non-end-block contains no cycle of length congruent to two modulo five. Every member of $\mathcal E_5$ has cycle-residue spectrum ${0,1,3,4}$. The proof combines structural arguments with finite computational checks. It uses the separately established Dean–5 theorem and its weak-graph strengthening as inputs. The contribution is the stronger stability classification; independent expert review remains pending.

The preprint claims an exact classification. Write $C_5(G)$ for the residues modulo five represented by cycle lengths in $G$. Let $\mathcal E_5={K_6,K_{5,5}}\cup{H_{5,n;t}:2\le t\le5<n}$, where $H_{5,n;t}$ is obtained from $K_{5,n}$ by deleting $5-t$ edges incident with one vertex in the part of size $n$. For every finite simple graph $G$ with minimum degree at least five, exactly one alternative holds: $C_5(G)=\mathbb Z_5$; or every end-block belongs to $\mathcal E_5$ and every non-end-block contains no cycle of length congruent to two modulo five. Every member of $\mathcal E_5$ has cycle-residue spectrum ${0,1,3,4}$. The proof combines structural arguments with finite computational checks. It uses the separately established Dean–5 theorem and its weak-graph strengthening as inputs. The contribution is the stronger stability classification; independent expert review remains pending.