84 On a Special Class of Hamiltonian Graphs GARY CHARTRAND 1) and HUDSON V. KRONK One of the most basic questions asked about a graph (finite, undirected, without loops or multiple edges) is whether its structure is such that it can be traversed or traced in a certain manner. Undoubtedly, the two most important classes of graphs dealing with traversability are the eulerian graphs and the hamiltonian graphs. A graph G is eulerian if it has a closed path (called an eulerian path) containing every edge of G exactly once and every vertex of G at least once, while G is hamiltonian if it has a closed path containing every vertex of G exactly once, i.e.. if it has a hamiltonian cycle. A graph G is said to be randomly eulerianfrom a vertex v if the following procedure always results in an eulerian path. Begin at the given vertex v and traverse any incident edge. On arriving at a vertex, choose any incident edge which has not yet been traversed. When no new edges are available the procedure terminates. These graphs have also been referred to as arbitrarily traversable from v and arbitrarily traceable from v and have been investigated by BABBLER [1], HARARY [3], and ORE [4]. This suggests the following concept. We define a graph G to be randomly hamil- tonian from the vertex v if the following procedure always results in a hamiltonian cycle. Begin at the vertex v and proceed to any adjacent vertex. On arriving at a vertex, select any adjacent vertex not previously encountered. When no new vertices remain, then an edge exists between the final vertex chosen and v, and the procedure terminates. Thus in a graph G which is randomly hamiltonian from a vertex v, any path beginning at v can be extended to a hamiltonian cycle. Graphs which are random- ly hamiltonian from every vertex were characterized in [2] and are called simply randomly hamiltonian graphs. It is the object of this article to present a characterization of graphs which are randomly hamiltonian from a vertex, and thereby provide a classification of all such graphs. It is convenient to introduce notation for several types of graphs which are encountered throughout the course of this article. The complete graph with p vertices is denoted by Kp, while Cp represents the cycle with p/> 3 vertices. The complete bipartite graph K(m, n) is the graph with p = m + n vertices whose vertex set V can be partitioned as V I w 1/2 such that IV 1I= m, [V2I= n, and vertices u and v are adjacent if and only if ue Vi and ve Vj, ir It was shown in [2] that a graph G with p 1>3 vertices is randomly hamiltonian if and only if it is one of the graphs Kp, C~, and K(p/2,p/2). t) Research supported in part by a Faculty Research Fellowship from Western Michigan University.