For a connected graph G of order n ≥ 3, let f : E(G) → Zn be an edge labeling of G. The vertex labeling f ′ : V (G) → Zn induced by f is defined as f (u) = ∑ v∈N(u) f(uv), where the sum is computed in Zn. If f ′ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A modular edge-graceful labeling f of G is nowhere-zero if f(e) 6= 0 for all e ∈...

Graceful labelings use a prominent place among difference vertex labelings. In this work we present new families of graceful graphs all of them obtained applying a general substitution result. This substitution is applied here to replace some paths with some trees with a more complex structures. Two caterpillars with the same size are said to be textit{analogous} if thelarger stable sets, in bo...

The Graceful Tree Conjecture claims that every finite simple tree of order n can be vertex labeled with integers {1, 2, ...n} so that the absolute values of the differences of the vertex labels of the end-vertices of edges are all distinct. That is, a graceful labeling of a tree is a vertex labeling f , a bijection f : V (Tn) −→ {1, 2, ...n}, that induces an edge labeling g(uv) = |f(u)− f(v)| t...

If the vertex set V of > =< V,E G can be divided into two un empty sets X and Y , ∅ = = Y V,X Y X ∩ ∪ , but also two nodes of every edge belong to X and Y separately, the G is called bipartite graph. If E ) ,y Y,(x X,y x i i i i ∈ ∈ ∈ ∀ then G is called complete bipartite graph. if n Y m, X = = , the G is marked m,n K . In this paper the graceful labeling, k-graceful labeling, odd graceful labe...

A graph G on m edges is considered graceful if there is a labelling f of the vertices of G with distinct integers in the set {0, 1, . . . ,m} such that the induced edge labelling g defined by g(uv) = |f(u) − f(v)| is a bijection to {1, . . . ,m}. We here consider some relaxations of these conditions as applied to tree labellings: 1. Edge-relaxed graceful labellings, in which repeated edge label...

vspace{0.2cm} Let $G$ be a graph and $f:V(G)rightarrow {1,2,3,.....left|V(G)right|}$ be a bijection. Let $p_{uv}=f(u)f(v)$ and\ $ d_{uv}= begin{cases} left[frac{f(u)}{f(v)}right] ~~if~~ f(u) geq f(v)\ \ left[frac{f(v)}{f(u)}right] ~~if~~ f(v) geq f(u)\ end{cases} $\ for all edge $uv in E(G)$. For each edge $uv$ assign the label $1$ if $gcd (p_{u...

This paper addresses the repeated acquisition of labels for data items when the labeling is imperfect. We examine the improvement (or lack thereof) in data quality via repeated labeling, and focus especially on the improvement of training labels for supervised induction. We show that the repeated-labeling strategies proposed acquire more labels for difficult examples automatically. Among these ...