Pullbacks and pushouts in the category of graphs

Question

Let $\textbf{Grph}$ be the category of simple, undirected graphs without loops, together with graph homomorphisms. Note that there need not be any homomorphisms between two graphs, for instance $\textrm{Hom}(G,K_2) = \emptyset$ for all graphs $G$ with $\chi(G) > 2$.

Does $\textbf{Grph}$ have pullbacks and pushouts?

Answer 1

Pullbacks exist in your $\mathbf{Grph}$ but pushouts do not always exist. For example, $$\require{AMScd}\begin{CD} \{ \bullet \quad \bullet \} @>>> \{ \bullet \} \\ @VVV \\ \{ \bullet - \bullet \} \end{CD}$$ has no pushout – indeed, no cocone exists.