On d-graceful labelings

In this paper we introduce a generalization of the well known concept of a graceful labeling. Given a graph G with e=dm edges, we call d-graceful labeling of G an injective function from V(G) to the set {0,1,2,..., d(m+1)-1} such that {|f(x)-f(y)| | [x,y]\in E(G)} ={1,2,3,...,d(m+1)-1}-{m+1,2(m+1),...,(d-1)(m+1)}. In the case of d=1 and of d=e we find the classical notion of a graceful labeling and of an odd graceful labeling, respectively. Also, we call d-graceful \alpha-labeling of a bipartite graph G a d-graceful labeling of G with the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other one. We show that these new concepts allow to obtain certain cyclic graph decompositions. We investigate the existence of d-graceful \alpha-labelings for several classes of bipartite graphs, completely solving the problem for paths and stars and giving partial results about cycles of even length and ladders.Comment: In press on Ars Combi

Alpha Labeling of Amalgamated Cycles

A graceful labeling of a bipartite graph is an \a-labeling if it has the property that the labels assigned to the vertices of one stable set of the graph are smaller than the labels assigned to the vertices of the other stable set. A concatenation of cycles is a connected graph formed by a collection of cycles, where each cycle shares at most either two vertices or two edges with other cycles in the collection. In this work we investigate the existence of \a-labelings for this kind of graphs, exploring the concepts of vertex amalgamation to produce a family of Eulerian graphs, and edge amalgamation to generate a family of outerplanar graphs. In addition, we determine the number of graphs obtained with $k$ copies of the cycle $C_n$, for both types of amalgamations

Prime Labelings of Snake Graphs

A prime labeling of a graph G with n vertices is a labeling of the vertices with distinct integers from the set {1, 2 ,..., n} such that the labels of any two adjacent vertices are relatively prime. In this paper, we introduce a snake graph, the fused union of identical cycles, and define a consecutive snake prime labeling for this new family of graphs. We characterize some snake graphs that have a consecutive snake prime labeling and then consider a variation of this labeling

New Results on Subtractive Magic Graphs

For any edge xy in a directed graph, the subtractive edge-weight is the sum of the label of xy and the label of y minus the label of x. Similarly, for any vertex z in a directed graph, the subtractive vertex-weight of z is the sum of the label of z and all edges directed into z and all the labels of edges that are directed away from z. A subtractive magic graph has every subtractive edge and vertex weight equal to some constant k. In this paper, we will discuss variations of subtractive magic labelings on directed graphs

On Integer Additive Set-Sequential Graphs

International audienceA set-labelling of a graph G is an injective function f : V (G) → P(X), where X is a finite set of non-negative integers and a set-indexer of G is a set-labeling such that the induced function f ⊕ : E(G) → P(X) − {∅} defined by f ⊕ (uv) = f (u)⊕f (v) for every uv∈E(G) is also injective. A set-indexer f : V (G) → P(X) is called a set-sequential labeling of G if f ⊕ (V (G) ∪ E(G)) = P(X) − {∅}. A graph G which admits set-sequential labelling is called a set-sequential graph. An integer additive set-labeling is an injective function f : V (G) → P(N 0 ), N 0 is the set of all non-negative integers and an integer additive set-indexer is an integer additive set-labeling such that the induced function f + : E(G) → P(N 0 ) defined by f + (uv) = f (u) + f (v) is also injective. In this paper, we extend the concepts of set-sequential labelling to integer additive set-labellings of graphs and provide some resultson them

K-product cordial labeling of fan graphs

Let f be a map from V (G) to {0, 1, ..., k − 1} where k is an integer, 1 ≤ k ≤ |V (G)|. For each edge uv assign the label f(u)f(v)(mod k). f is called a k-product cordial labeling if |vf (i) − vf (j)| ≤ 1, and |ef (i) − ef (j)| ≤ 1, i, j ∈ {0, 1, ..., k − 1}, where vf (x) and ef (x) denote the number of vertices and edges respectively labeled with x (x = 0, 1, ..., k − 1). In this paper we prove that fan Fn and double fan DFn when k=4 and 5 admit k-product cordial labeling.Publisher's Versio

Discussion on some interesting topics in Graph Theory

Not availabl