Edge-graceful labeling

In graph theory, an edge-graceful graph labeling is a type of graph labeling. This is a labeling for simple graphs in which no two distinct edges connect the same two distinct vertices, no edge connects a vertex to itself, and the graph is connected. Edge-graceful labelings were first introduced by S. Lo in his seminal paper. In graph theory, an edge-graceful graph labeling is a type of graph labeling. This is a labeling for simple graphs in which no two distinct edges connect the same two distinct vertices, no edge connects a vertex to itself, and the graph is connected. Edge-graceful labelings were first introduced by S. Lo in his seminal paper. Given a graph G, we denote the set of edges by E(G) and the vertices by V(G). Let q be the cardinality of E(G) and p be that of V(G). Once a labeling of the edges is given, a vertex u of the graph is labeled by the sum of the labels of the edges incident to it, modulo p. Or, in symbols, the induced labeling on the vertex u is given by where V(u) is the label for the vertex and E(e) is the assigned value of an edge incident to u. The problem is to find a labeling for the edges such that all the labels from 1 to q are used once and the induced labels on the vertices run from 0 to p − 1. In other words, the resulting set for labels of the edges should be { 1 , 2 … q } {displaystyle {1,2dots q}} and { 0 , 1 … p − 1 } {displaystyle {0,1dots p-1}} for the vertices. A graph G is said to be edge-graceful if it admits an edge-graceful labeling. Consider a path with two vertices, P2. Here the only possibility is to label the only edge in the graph 1. The induced labeling on the two vertices are both 1. So P2 is not edge-graceful. Appending an edge and a vertex to P2 gives P3, the path with three vertices. Denote the vertices by v1, v2, and v3. Label the two edges in the following way: the edge (v1, v2) is labeled 1 and (v2, v3) labeled 2. The induced labelings on v1, v2, and v3 are then 1, 0, and 2 respectively. This is an edge-graceful labeling and so P3 is edge-graceful. Similarly, one can check that P4 is not edge-graceful. In general, Pm is edge-graceful when m is odd and not edge-graceful when it is even. This follows from a necessary condition for edge-gracefulness (see below).