Edge-graceful labeling
Encyclopedia
In graph theory
, an edge-graceful graph labeling is a type of graph labeling
. This is a labeling for simple graphs, namely ones 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.
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
and 
for the vertices.
A graph G is said to be edge-graceful if it admits an edge-graceful labeling.
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).
with three vertices, C3. This is simply a triangle. One can label the edges 0, 1, and 2, and check directly that, along with the induced labeling on the vertices, this gives an edge-graceful labeling.
Similar to paths,
is edge-graceful when m is odd and not when m is even.
An edge-graceful labeling of
is shown in the following figure:
is congruent to 
modulo p.
or, in symbols,
This is referred to as Lo's condition in the literature. This follows from the fact that the sum of the labels of the vertices is twice the sum of the edges, modulo p. This is useful for disproving a graph is edge-graceful. For instance, one can apply this directly to the path and cycle examples given above.
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, an edge-graceful graph labeling is a type of graph labeling
Graph labeling
In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to the edges or vertices, or both, of a graph....
. This is a labeling for simple graphs, namely ones 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.
Definition
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 bywhere 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
and for the vertices.A graph G is said to be edge-graceful if it admits an edge-graceful labeling.
Paths
Consider a pathPath (graph theory)
In graph theory, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the next vertex in the sequence. A path may be infinite, but a finite path always has a first vertex, called its start vertex, and a last vertex, called its end vertex. Both of them...
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).
Cycles
Consider the cycleCycle graph
In graph theory, a cycle graph or circular graph is a graph that consists of a single cycle, or in other words, some number of vertices connected in a closed chain. The cycle graph with n vertices is called Cn...
with three vertices, C3. This is simply a triangle. One can label the edges 0, 1, and 2, and check directly that, along with the induced labeling on the vertices, this gives an edge-graceful labeling.
Similar to paths,
is edge-graceful when m is odd and not when m is even.An edge-graceful labeling of
is shown in the following figure:A necessary condition
Lo gave a necessary condition for a graph to be edge-graceful. It is that a graph with q edges and p vertices is edge graceful only if is congruent to modulo p.or, in symbols,
This is referred to as Lo's condition in the literature. This follows from the fact that the sum of the labels of the vertices is twice the sum of the edges, modulo p. This is useful for disproving a graph is edge-graceful. For instance, one can apply this directly to the path and cycle examples given above.
Further selected results
- The Petersen graphPetersen graphIn the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named for Julius Petersen, who in 1898 constructed it...
is not edge-graceful.
- The star graph
(a central node and m legs of length 1) is edge-graceful when m is even and not when m is odd.
- The friendship graphFriendship graphIn the mathematical field of graph theory, the friendship graph Fn is a planar undirected graph with 2n+1 vertices and 3n edges....
is edge-graceful when m is odd and not when it is even.
- Regular trees,
(depth n with each non-leaf node emitting m new vertices) are edge-graceful when m is even for any value n but not edge-graceful whenever m is odd.
- The complete graphComplete graphIn the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.-Properties:...
on n vertices,
, is edge-graceful unless n is singly even, 
.
- The ladder graphLadder graphIn the mathematical field of graph theory, the ladder graph Ln is a planar undirected graph with 2n vertices and n+2 edges....
is never edge-graceful.