Did my math textbook make a typo?

Question

My textbook defined connectedness in graphs in the following way:

A graph G(V, E) is said to be connected if for every pair of vertices u and v there is a path in G from u to v.

The textbook then asks the reader to complete the following exercise:

Show by giving an example that there are connected graphs in which there is no path that goes through all the vertices.

I take the exercise as impossible to complete. By definition, connected graphs are those in which there is some path that goes through all the vertices. Might I have misunderstood either the definition or exercise? Could it be possible that the textbook made a typo?

Any advice or clarification would be greatly appreciated.

Answer 1

A connected graph means that between any two vertices, there is some path. For example if the two vertices share an edge, there is a path between them consisting of just that single edge.

This says nothing about whether or not there is a single path that goes through all the vertices in some order. Remember that by definition a path cannot visit any vertex more than once. So as you go from one vertex to the next, it might not be possible to visit all vertices without re-visiting a previous one.

Answer 2

The easiest way to explain may be to provide a solution to the question. Consider the graph on $4$ vertices given by $$G=(\{1,2,3,4\},\{(1,2),(1,3),(1,4)\})$$ Then for every pair of vertices there is a path between them (so $G$ is connected) but there is no path connecting every vertex.