Eulerian and Hamiltonian Graphs · Hard problem

A graph has a clique on $\{1,2,3,4,5\}$ (all $10$ edges present), plus vertex $6$ joined to $3,4,5$, plus vertex $7$ joined to $1,2,4$ ($7$ vertices total). Among all non-adjacent pairs, find the smallest value of $\deg(u)+\deg(v)$, and state whether Ore's theorem guarantees a Hamiltonian cycle here.