2 problems

A graph pair $(\Gamma,\Sigma)$ is nontrivial if $\Gamma$ and $\Sigma$ are coprime connected twin-free graphs and exactly one of them is bipartite. A graph pair is stable if it has…

Let $K[a_i,b_i]$ denote the complete $b_i$-partite graph with $a_i$ vertices in each part, and let $G$ be its direct product … Assume $2\leq b_1\leq b_2\leq\cdots\leq b_t$. The upp…