Spanning trails

For a graph G with distinguished vertices u and u, w e give a sufficient condition for the existence of a ( u , u)-trail containing every vertex of G. In this paper we follow the notation of Bondy and Murty 11, except that the graph G is simple with n vertices and m edges. For 1 4 , u E V ( G ) . a (u . u)-trnil is a sequence xu, e , , x,, e2 . . . . , .rT , , e , , x, whose terms are alternately vertices and edges, with e, joining x , and x, ( I 5 i 5 s), where the edges are distinct. and where 14 = x,) is the origin and u = x, is the terminus. A (u,u)-trail spans G if i t contains every vertex of G, and it is closed if I I = u. We denote by d ( u ) the degree of u in G and by dH(u ) the degree of u in the subgraph H . The neighborhood of u, denoted N ( u ) , is the set of vertices adjacent to u. We shall prove the following result: Theorem 1. I I , U E V ( G ) . I f Let G be a graph on IZ vertices, with no vertex isolated, and let for each edge xy E E(G), then exactly one of the following holds: ( i ) G has a spanning ( u , u)-trail. ( i i ) d(z) = 1 for some vertex z {u ,u } . ( i i i ) G = K? ,,,?. u = u , and n is odd. ( iv) G = K , ,,,?, u # u. uu @ E ( G ) , n is even, and d(u) = d(u) = i? 2. (v) I I = u, and u is the only vertex with degree 1 in G. Theorem 1 is rnotivatcd by some recent results on Hamiltonian line graphs. Harary and Nash-Williams [ 5 ] gave this characterization: Theorem 2 (Harary and Nash-Williams). Let G be a graph with at least 4 vertices. The line graph L ( G ) is Hamiltonian if and only if G has a closed trail that contains at least one vertex of each edge of G. Journal of Graph Theory, Vol. 11, No. 2, 161-167 (1987)