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...

An injective function f : V pGq Ñ t0, 1, 2, . . . , qu is an odd sum labeling if the induced edge labeling f defined by f puvq fpuq fpvq, for all uv P EpGq, is bijective and f pEpGqq t1, 3, 5, . . . , 2q 1u. A graph is said to be an odd sum graph if it admits an odd sum labeling. In this paper we study the odd sum property of graphs obtained by duplicating any edge of some graphs.

Let G be a (p, q) graph and let f : V (G) → {1, 2, 3, · · · , p + q} be an injection. For each edge e = uv, let f∗(e) = (f(u)+f(v))/2 if f(u)+f(v) is even and f∗(e) = (f(u)+f(v)+1)/2 if f(u) + f(v) is odd. Then f is called a super mean labeling if f(V ) ∪ {f∗(e) : e ∈ E(G)} = {1, 2, 3, · · · , p+ q}. A graph that admits a super mean labeling is called a super mean graph. In this paper we presen...

Cavenagh and Wanless recently proved that, for sufficiently large odd n, the number of transversals in the Latin square formed from the addition table for integers modulo n is greater than (3.246). We adapt their proof to show that for sufficiently large t the number of additive permutations on [−t, t] is greater than (3.246) and we go on to derive some much improved lower bounds on the numbers...

We conjecture that any 2-regular simple graph has an SSA labeling. We provide several special cases to support our conjecture. Most of our constructions are based on Skolem sequences and on an extension of it. We establish a connection between simply sequentially additive labelings of 2-regular graphs and ordered graceful labelings of spiders.