On sequential labelings of graphs

A valuation on a simple graph G is an assignment of labels to the vertices of G which induces an assignment of labels to the edges of G. β‐valuations, also called graceful labelings, and α‐valuations, a subclass of graceful labelings, have an extensive literature; harmonious labelings have been introduced recently by Graham and Sloane. This paper introduces sequential labelings, a subclass of harmonious labelings, and shows that any tree admitting an α‐valuation also admits a sequential labeling and hence is harmonious. Constructions are given for new families of graceful and sequential graphs, generalizing some earlier results. Finally, a conjecture of Frucht is shown to be wrong by exhibiting several graceful labelings of wheels in which the center label is larger than previously thought possible.