A k-extended Skolem-type 5-tuple difference set of order t is a set of t 5-tuples {(di,1, di,2, di,3, di,4, di,5) | i = 1, 2, . . . , t} such that di,1+di,2+di,3+di,4+di,5 = 0 for 1 ≤ i ≤ t and {|di,j| | 1 ≤ i ≤ t, 1 ≤ j ≤ 5} = {1, 2, . . . , 5t+1}\{k}. In this talk, we will give necessary and sufficient conditions on t and k for the existence of a k-extended Skolem-type 5-tuple difference set ...

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...

A difference vertex labeling of a graph G is an assignment f of labels to the vertices of G that induces for each edge xy the weight |f(x)− f(y)| . A difference vertex labeling f of a graph G of size n is odd-graceful if f is an injection from V (G) to {0, 1, ..., 2n − 1} such that the induced weights are {1, 3, ..., 2n − 1}. We show here that any forest whose components are caterpillars is odd...

Edge (5, 3)-colorings of a (5, 3)-regular bipartite graph relating 5cycles and vertices of the Petersen graph lead to solutions of the Great Circle Challenge. This takes, for odd n = 2k + 1 > 1, to a canonical 1-1 correspondence from the family of n-cycles of the complete graph Kn onto the family Ck of n-cycles of the odd graph Ok+1 and onto the family of their pullback 2n-cycles in the middle-...

An approach to edge bicoloring problems for biregular bipartite graphs is applied to graphs arising from cyclic groups as well as from odd graphs. In the case of the Petersen graph, this incidentally allows solutions to the Great Circle Challenge puzzle. On the other hand, for odd n = 2k+1 > 1, a one-to-one correspondence from the family of n-cycles of Kn onto the family Ok of n-cycles of the o...

In this paper, we show that the class of Extended Duplicate Graph of a Twig is Graceful and Skolem-graceful. AMS SUBJECT CLASSIFICATION: 05C78.

Cyclic m-cycle systems of order v are constructed for all m ≥ 3, and all v ≡ 1(mod 2m). This result has been settled previously by several authors. In this paper, we provide a different solution, as a consequence of a more general result, which handles all cases using similar methods and which also allows us to prove necessary and sufficient conditions for the existence of a cyclic m-cycle syst...