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...

Graceful labeling is one of the best known labeling methods of graphs. Despite the large number of papers published on the subject of graph labeling, there are few particular techniques to be used by researchers to gracefully label graphs. In this paper, first a new approach based on the mathematical programming technique is presented to model the graceful labeling problem. Then a “branching me...

The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also show that the subdivision of ladder graph Ln with m-pendant S(Ln)  mk1 is odd grace...

A function f is called an odd-even graceful labeling of a graph G if f: V(G) → {0,1,2,...,q} is injective and the induced function f : E(G) → { { 0,2,4,...,2q+2i/i= 1 to n} such that when each edge uv is assigned the label |f(u) – f(v)| the resulting edge labels are {2,4,6,...,2q}. A graph which admits an odd-even graceful labeling is called an odd-even graceful graph. In this paper, the odd-ev...

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-...

Combinatorics of complexes plays an important role in topology, nonlinear analysis, game theory, and mathematical economics. In 1967, Ky Fan used door-to-door principle to prove a combinatorial Stokes’ theorem on pseudomanifolds. In 1993, Shih and Lee developed the geometric context of general position maps, π-balanced and π-subbalanced sets and used them to prove a combinatorial formula for mu...