Skolem-labeling of generalized three-vane windmills

A graph on 2n vertices can be Skolem-labeled if the vertices can be given labels from {1, . . . , n} such that each label i is assigned to exactly two vertices and these vertices are at distance i. Mendelsohn and Shalaby have characterized the Skolem-labeled paths, cycles and windmills (of fixed vane length). In this paper, we obtain necessary conditions for the Skolem-labeling of generalized k-windmills in which the vanes may be of different length. We show that these conditions are sufficient in the case where k = 3 and conjecture that any generalized k-windmill, k > 3, can be Skolem-labeled if and only if it satisfies these necessary conditions.