The purpose of the paper is to study relations graphs and certain Skolem sequences.
between graceful numbering of certain 2-regular
In this paper, all graphs will be finite, without loops or multiple edges. For any graph G, the symbols V(G) and E(G) will denote its vertex set and its edge set, respectively. A graceful numbering of a graph G with m vertices and n edges is a one-to-one mapping q of the set V(G) into the set (0, 1, . . . , n } which has the property that the values of the edges form the set {I, 2, . . . , n} if the value g(e) of the edge e with the end vertices u, u is defined by g(e) = Iv(u) -r/~(u)l. A graph is called graceful if it has a graceful numbering. An cu-valuation 1~ of a graph G is a graceful numbering of G which satisfies the following additional condition: There exists a number r (OS r s ]E(G)]) such that for any edge e = (v, w), min(WJ)* V(w)) s r < max(V(u), q(w)). The concepts of a graceful numbering and of an a-valuation were introduced by Rosa [6); in his paper as well as in [3] and [4]. the term '/?-valuation' was used for graceful numbering.
It should be noted that a graph with an cu-valuation is always bipartite. In this paper, ' * similarly as in some of the papers auoted, a gra$ G pi!! be called Eulerian if lE(G j] i ii and if every vertex of G is of even degree. Rosa proved that any graceful Eulerian graph G satisfies the condition IE(G)I = 0 or 3 (mod 4). This implies (see Kotzig [3, Theorem 2)) that any Eulerian gracefu! bipartite graph satisfies the condition IE(G)) = 0 (mod 4); in particular. any Eulerian graph with an a-valuation satisfies this condition.