4. Recall that an Euler circuit for G is a sequence of adjacent vertices and edges in G that starts and ends at the same vertex, uses every vertex of G at least once, and uses every edge of G exactly once. An Euler path from v to w in G is a sequence of adjacent edges and vertices that starts at v, ends at w, passes through every vertex of G at least once, and traverses every edge of G exactly once.
To determine whether the given graphs contain Euler circuits or Euler paths, use Theorem 11.2.4 and Corollary 11.2.5. Theorem 11.2.4 states that a nonempty graph G has an Euler circuit if, and only if, G is connected and every vertex of G has has even degree. Corollary 11.2.5 states that that there is an Euler path from v to w in G if, and only if, G is connected, v and w have odd degree, and all other vertices have even degree.