Given an undirected graph G, an L(h, k)-labelling of G assigns colors to vertices from the integer set {0, . . . , λh,k}, such that any two vertices vi and vj receive colors c(vi) and c(vj) satisfying the following conditions: i) if vi and vj are adjacent then |c(vi)− c(vj)| ≥ h; ii) if vi and vj are at distance two then |c(vi)− c(vj)| ≥ k. The aim of the L(h, k)-labelling problem is to minimiz...

Graceful labelling is studied on undirected graphs since graceful graphs can be used in some H-decomposition problems. In this note, we investigate the directed graceful problem for many orientations of undirected trees with short diameters, and provide some directed trees that deny any digraceful labelling. AMS Subject Classification (2000): 05C78

(V,E) be a finite undirected graph with vertex set V and edge set E. A mapping f ~ {O.l}m is called an M-coding of G, if the induced mapping g : V ~ {O,l • defined as g(v) f(uv). assigns ueV,uveE different vectors to the vertices of G, where all summations are mod 2.An M-coding f is calle~ positive if the zero vector is not assigned to any edge and the induced labelling g does not assign the ze...

We show that there exists an adjacency labelling scheme for planar graphs where each vertex of n -vertex graph G is assigned a (1 + o(1)) log 2 -bit label and the labels two vertices u v are sufficient to determine if uv edge . This optimal up lower order term first such asymptotically result. An alternative, but equivalent, interpretation this result that, every positive integer , U with 1+o(1...

For positive integers k and d ≥ 2, a k-S(d, 1)-labelling of a graph G is a function on the vertex set of G, f : V (G) → {0, 1, 2, · · · , k− 1}, such that |f(u)− f(v)|k ≥ { d if dG(u, v) = 1; 1 if dG(u, v) = 2, where |x|k = min{|x|, k − |x|} is the circular difference modulo k. In general, this kind of labelling is called the S(d, 1)-labelling. The σdnumber of G, σd(G), is the minimum k of a k-...