Let K_5 be the complete graph on five vertices. Complete-graph K_5 conjecture. H^1,3m-2_ A_m(K_5)= Z_2 oplusZ^5,&m=2;Z_2 oplusZ_3^4oplusZ^10,&m=3;Z_2^5oplusZ_m^10oplusZ^10,&m=4;Z_2^5oplusZ_m^10oplusZ^10,&m>4, m odd;Z_2^10oplusZ_m^10oplusZ^10,&m>5, m even. The source presents this as a conjectural computational pattern and gives no resolution.

Cohomology conjecture for the complete graph (K_5)

Let $K_5$ be the complete graph on five vertices. Complete-graph $K_5$ conjecture.

H^{1,3m-2}_{{\mathcal A}_m}(K_5)=\begin{cases}\mathbb{Z}_2\oplus\mathbb{Z}^{5},&m=2;\\mathbb{Z}_2\oplus\mathbb{Z}_3^{4}\oplus\mathbb{Z}^{10},&m=3;\\mathbb{Z}_2^{5}\oplus\mathbb{Z}_m^{10}\oplus\mathbb{Z}^{10},&m=4;\\mathbb{Z}_2^{5}\oplus\mathbb{Z}_m^{10}\oplus\mathbb{Z}^{10},&m>4,\ m\text{ odd};\\mathbb{Z}_2^{10}\oplus\mathbb{Z}_m^{10}\oplus\mathbb{Z}^{10},&m>5,\ m\text{ even}. \end{cases}

The source presents this as a conjectural computational pattern and gives no resolution.