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.
Please note that there will be more than one correct answer for the circuits and paths.
Graph (I) has an Euler circuit: a,c,b,e,b,a,e,c,d,e,d,a.
Graph (II) has an Euler circuit: a,b,d,e,b,c,f,e,h,f,i,h,g,d,a.
Graph (III) does not have an Euler circuit, but does have an Euler path (since there are exactly two vertices of odd degree): c,b,a,d,c,e,d,b,f,a,e,f.
Graph (IV) does not have an Euler circuit and does not have an Euler path.