2 problems

Let $G$ be a bipartite graph, let $d_u$ denote the degree of $u\in V(G)$, and let $H$ be a loop-graph. Write $\hom(G,H)$ for the number of graph homomorphisms from $G$ to $H$. Galv…

Existence conjecture. For every connected graph $F$ of order $3$ or more, there exists a non-trivial $F$-irregular graph.