Edge-graceful labeling

In graph theory, an edge-graceful labeling is a type of graph labeling for simple, connected graphs in which no two distinct edges connect the same two distinct vertices and no edge connects a vertex to itself.

Edge-graceful labelings were first introduced by Sheng-Ping Lo in his seminal paper.

Definition

Given a graph Template:Mvar, we denote the set of its edges by Template:Math and that of its vertices by Template:Math. Let Template:Mvar be the cardinality of Template:Math and Template:Mvar be that of Template:Math. Once a labeling of the edges is given, a vertex of the graph is labeled by the sum of the labels of the edges incident to it, modulo Template:Mvar. Or, in symbols, the induced labeling on a vertex is given by

$V(u)=\mathrm{\Sigma }E(e)mod|V(G)|$

where Template:Math is the resulting value for the vertex Template:Mvar and Template:Math is the existing value of an edge Template:Mvar incident to Template:Mvar.

The problem is to find a labeling for the edges such that all the labels from Template:Math to Template:Mvar are used once and that the induced labels on the vertices run from Template:Math to Template:Math. In other words, the resulting set of labels for the edges should be Template:Math, each value being used once, and that for the vertices should be Template:Math.

A graph Template:Mvar is said to be edge-graceful if it admits an edge-graceful labeling.

Examples

Cycles

An edge-graceful labeling of Template:Math

Consider the cycle with three vertices, Template:Math. This is simply a triangle. One can label the edges 1, 2, and 3, and check directly that, along with the induced labeling on the vertices, this gives an edge-graceful labeling. Similar to paths, Template:Mvar is edge-graceful when Template:Mvar is odd and not when Template:Mvar is even.

Paths

Consider a path with two vertices, Template:Math. 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 Template:Math is not edge-graceful.

Appending an edge and a vertex to Template:Math gives Template:Math, the path with three vertices. Denote the vertices by Template:Math, Template:Math, and Template:Math. Label the two edges in the following way: the edge Template:Math is labeled 1 and Template:Math labeled 2. The induced labelings on Template:Math, Template:Math, and Template:Math are then 1, 0, and 2 respectively. This is an edge-graceful labeling and so Template:Math is edge-graceful.

Similarly, one can check that Template:Math is not edge-graceful.

In general, Template:Mvar is edge-graceful when Template:Mvar is odd and not edge-graceful when it is even. This follows from a necessary condition for edge-gracefulness.

A necessary condition

Lo gave a necessary condition for a graph with Template:Mvar edges and Template:Mvar vertices to be edge-graceful:Template:Math must be congruent to Template:Math. In symbols:

$q(q+1)\equiv \frac{p(p-1)}{2}modp.$

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 Template:Mvar. 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 graph is not edge-graceful.
• The star graph $S_m$ (a central node and m legs of length 1) is edge-graceful when m is even and not when m is odd.
• The friendship graph $F_m$ is edge-graceful when m is odd and not when it is even.
• Regular trees, $T_{m,n}$ (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 graph on n vertices, $K_n$, is edge-graceful unless n is singly even, $n=2mod4$.
• The ladder graph is never edge-graceful.