On The Edge-Graceful and
Edge-Magic Maximal
Outerplanar Graphs
Sin-Min Lee , Medei Kitagaki,
Joseph Young
Department of Computer Science
San Jose State University
San Jose, California 95192 U.S.A.
William Kocay
Department of Computer Science
University of Manitoba
Winnipeg, Canada R3T 2N2
Definition of edge-graceful
 Definition 1 An edge-graceful labeling of a
graph G with p vertices and q edges is a one-toone correspondence f from E(G) to {1, 2, . . ., q}
such that for all vertices v, the vertex sums are
distinct, mod p.
A graph G is called edge-graceful if there exists
an edge-graceful labeling of G.
Concept Introduced by: Lo Sheng -Ping
A necessary condition of edge-gracefulness is:
q(q+1)  p(p-1)/2(mod p)
This condition may be more practically stated as
q(q+1)  0 or p/2 (mod p) depending on whether p is odd
or even.
Lee proposed the following tantalizing conjectures:
Conjecture 1: The Lo condition (2) is sufficient for a connected
graph to be edge-graceful.
A sub-conjecture of the above that has not yet been proved:
Conjecture 2: All odd-order trees are edge-graceful.
Lemma Let G be a maximal outerplanar graph
with n vertices, n≥3, then
(i) there are 2n-3 edges, in which there are n-3
chords;
(ii) there are n-2 inner faces. Each inner face is
triangular;
(iii) there are at least two vertices with degree 2;
Theorem. A maximal outerplanar graph with p vertices is edgegraceful if p = 4 and 12.
Proof. A maximal outerplanar graph with p vertices has 2p-3 edges.
It is edge-graceful if it satisfies the Lo’s condition
q(q+1)  p(p-1)/2 (mod p)
 (2p-3)(2p-2)  p(p-1)/2 (mod p)
 (4p-6)(p-1)  p(p-1)/2 (mod p)
 4p-6  p/2 (mod p)
Thus p is even, say p = 2k. We have 7k  6 (mod 2k),
i.e. k  6 (mod 2k).
From which we have k = 2 or k = 6. Thus p = 4 or 12.
Theorem. The maximal outerplanar graph with 4 vertices is
edge-graceful.
Our true task: To show there exists an edgegraceful labeling for every maximal
outerplanar graph with p = 12 vertices.
Bob – There is only 1 unique graph for p = 4, but
HOW MANY ARE THERE FOR p = 12?
Alice – Just draw every single graph and count.
Bob - HOW DO WE ENSURE WE HAVE ALL THE
GRAPHS?
GENERATING MAXIMAL OUTERPLANAR GRAPHS
“The Idea”
Notice that given any MOP with n vertices, we can
construct a new MOP with n + 1 vertices by simply adding
a triangle on one of the outer edges.
n=6
n=7
GENERATING MAXIMAL OUTERPLANAR GRAPHS
“The Algorithm Skeleton”
While(n < 12)
{
While(more graphs of n vertices)
While(i < n - 1)
{
Add new vertex connected to vertex i and i+1
increment i
Store new graph of n+1
}
Add a new vertex connecting vertex 1 and n
Store new graph of n+1
}
*Sort out duplicates from stored set of graphs of n+1
}
*The algorithm generates many unnecessary,
isomorphic graphs.
At each stage in the generation process - for each
set of MOP's with n vertices generated - we must
some how sort out and eliminate duplicates.
Professor William Kocay's 'Groups and
Graphs' software exploits Automorphism to
sort out from a set of graphs all the duplicates.
GENERATING MAXIMAL OUTERPLANAR GRAPHS
“The Algorithm Revised”
While(n < 12)
{
While(more graphs of n vertices)
While(i < n - 1)
{
Add a new vertex connected to vertex i and i+1
increment i
Store new graph of n+1
}
Add a new vertex connecting vertex 1 and n
Store new graph of n+1
}
*Feed graphs through 'Groups and Graphs'
}
Now that we have all the graphs we needed, we must find an
edge-graceful labeling for each graph.
-To find edge-graceful labelings, we search through every
permutation of edge-weight labels.
-For MOP's with 12 vertices, there are 21 edges.
-There are 21! Possible ways to label(weigh) the edges.
-There are 733 MOP's with 12 vertices.
CONCLUSION: That is a lot of permutations.
The task of finding edge-graceful labels becomes a
problem of complexity.
To minimize overhead time, we use the
iterative version of Sedgewick's heap
method algorithm to generate
permutations.
To test every permutation sequentially is
unsound, considering the following
empirical result:
Expected time to find a labeling for a graph:
About 3 days.
(There are 733 graphs!)
Empirical result was inferred from a small
data set - some graphs we ran sequentially
- but a better method is desirable.
Solution: Exploit the nature of our software
-The order in which the edges are input
matters.
-If an edge-graceful label is not found after
a few thousand permutations, randomize the
input and try again.
Theorem. The maximal outerplanar graph with 12 vertices are
edge-graceful.
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M14
The dual concept of edge-graceful graphs was
introduced in 1992.
Let G be a (p,q) graph in which the edges are
labeled 1,2,3,...q so that the vertex sums are constant,
mod p. Then G is said to be edge-magic..
Sin-Min Lee, E. Seah and S.K. Tan , On
edge-magic graphs, Congressus
Numerantium 86 (1992), 179-191.
A necessary condition for a (p,q)-graph to be edgemagic is q(q+1) 0 (mod p).
However, this condition is not sufficient.
Examples:
(1) Trees
(2) Cycles
Theorem. A maximal outerplanar graph with p vertices
is edge-magic if p= 6.
Proof. A maximal outerplanar graph with p vertices is
edge-magic if it satisfies
q(q+1)0 (mod p)
 (2p-3)(2p-2) 0 (mod p)
 (4p-6)(p-1) 0 (mod p)
 4p-6 0(mod p)
 6 0(mod p)
Thus p is 6. 
Theorem . All maximal outerplanar graph with 6 vertices are edge-magic.
Proof. Up to isomorphism there are three maximal outerplanar graphs of order
6.