Find minimal spanning tree of graph with edge values from 1 to 5 integers
Take a look at the complexity of Kruskal's algorithm. Since the edges are with integer weights, you can sort them in linear time. The entire algorithm will take a total of $O(|V|\alpha(|E|)+|E|)$. We need $O(|E|)$ to sort, and an additional $O(|V|\alpha(|E|))$ for Kruskal's algorithm.