Graceful Graphs

Kari Lock
Wll Williams College
Hudson River Undergraduate
Mathematics Conference 2003

Dfi Definition ii
Definition:Agraceful labeling is a
labeling of the vertices of a graph with
distinct integers from the set {0, 1, 2, ... ,
q} (where q represents the number of
edges) such that...
if f(v) denotes the label even to vertex v, when
each edge uv is given the value | f(u) – f(v) |,
the edges are labeled 1, 2, ... , q

Example: K 3
0
2 3
2 3
1

Dfi Definition ii
Definition: A graph G is graceful if
and only if...
G can be labeled gracefully.

Are The Following Graphs Graceful?
• Star Graphs?
• Path Graphs?
•Cycle C l Graphs?
• Complete Graphs?
• Complete Bipartite Graphs?
• Wheel Graphs?
• Polyhedral Graphs?
• Trees???

Star Graphs
1 2
1 2
7 3
7 3
0
6 4
5
6 4
5
Theorem: Every star graph his graceful.

Path Graphs
Theorem: Every path graph his graceful.

Proof: Let G be a path graph.
Path Graphs
• Label the first vertex 0, and label every other vertex
increasing by 1 each time.
• Label the second vertex q and label every other vertex
decreasing by 1 each time.
• There are q + 1 vertices, so the first set will label l it’s
vertices with numbers from the set
• {0 1 q / 2} if q is even and from the set {0 1
• {0, 1, ... , q / 2} if q is even and from the set {0, 1, ... ,
(q+1)/2} if q is odd. The second set will label it’s vertices
with numbers from the set {(q+2)/2, ... , q} if q is even, and
{(q+3)/2, ... , q} if q is odd. Thus, the vertices are labeled
legally.

Path Graphs
• With the vertices labeled in this manner, the edges
attain the values q, q-1, q-2, ... 1, in that order.
•Thus, this is a graceful labeling, so G is graceful.
•Therefore, all path graphs are graceful.

Path Graphs
3
0
3
2
1
1
2
Theorem: Every path graph his graceful.

0
Cycle Graphs
2 3
=> NOT GRACEFUL
2
1
3
0 3 3
4
1
4 2
2
Theorem:C p is graceful if and only if 4|p or 4|(p+1)

Eulerian Graphs
Theorem: If G is a (p, q) graceful Eulerian
graph, then 4|q or 4|(q+1).

Complete Graphs
0 1 1
0
0 2 2
2 3
4
6
3
5
2
1
3
6 1 5
Theorem: K2, K3, K4 are the only graceful
complete graphs.

More Graceful Graphs
‣ Complete Bipartite Graphs
‣ Wheel Graphs
‣ Polyhedral Graphs
‣ Peterson Graph
‣ All graphs of order 4 or less
‣ All graphs of order 5 except...

More Graceful Graphs
‣ Trees???

Tree Example
Def: A tree is a connected graph with no cycles

Trees
Kotzig’s Conjecture: Every nontrivial tree is graceful.
This has been proved for p less than or equal
to 16, and is generally assumed to be true for
all trees, but no one can prove it!
=> BIG QUESTION FOR GRACEFUL
=> BIG QUESTION FOR GRACEFUL
GRAPHS: IS EVERY TREE GRACEFUL???

Definition of Graceful???
Def: A graceful labeling is a labeling of the vertices of a graph
with distinct integers from the set {0, 1, 2, ... , q} (where q is
the number of edges) such that when each edge uv is given the
value | f(u) – f(v) |, the edges are labeled 1, 2, ... , q
• integers from the set {0, 1, 2, ... , q}
• integers
• nonnegative integers
Maybe they
are all the
same!!!
• positive integers ???
OH NO!

Conjecture 1
Conjecture 1: If a graph G can be gracefully
labeled by labeling the vertices from the set of
integers, then G can be gracefully labeled by
labeling the vertices from the set of nonnegative
integers.

Conjecture 1
Proof: Let G be a gracefully labeled graph, with the vertices labeled
from the set of all integers.
Call the smallest integer k.
Subtract k from every vertex labeling.
The smallest vertex labeling now is k – k = 0, so all vertices are
labeled with nonnegative integers.
For any two vertices u, v є V(G), the edge uv originally had the value
| f(u) – f(v) |.
The edge uv now has value | (f(u) – k – (f(v) – k) |
= | f(u) – k – f(v) + k | = | f(u) – f(v) |.
Thus, the edge values are preserved so this is still a graceful labeling.
li

Theorem 1
Theorem 1: If a graph G can be gracefully
g p g y
labeled by labeling the vertices from the set of
integers, then G can be gracefully labeled by
labeling the vertices from the set of nonnegative
integers.

Conjecture 2
Conjecture 2: If a graph G can be gracefully
labeled by labeling the vertices from the set of
integers, then G can be gracefully labeled by
labeling the vertices from the set of positive
integers.

Theorem 2
Theorem 2: If a graph G can be gracefully
labeled by labeling the vertices from the set of
integers, then G can be gracefully labeled by
labeling the vertices from the set of positive
integers.

Definition i i of Graceful???
Def: A graceful labeling is a labeling of the vertices of a
graph with distinct integers such that when each edge uv is
given the value | u-v |, the edges are labeled 0, 1, 2, ... , q
(where q is the number of edges).
• integers
• nonnegative integers
• positive integers
• integers from the set {0, 1, 2, ... , q}
INTERCHANGEABLE
IN THE DEFINITION!

Conjecture 3
Conjecture 3: If a (p,q) graph G can be
gracefully labeled by labeling the vertices from
the set of integers, then G can be gracefully
labeled by labeling the vertices from the set
{0, 1, 2, ... , q}.
Unfortunately, this is still a conjecture.

Importance of Conjecture 3
If Conjecture 3 is true, I will be able to prove
that all trees are graceful!!!
Conjecture 4: If the fact that a (p,q) graph G can
be gracefully fll labeled lbldby lbli labeling the vertices from
the set of integers implies that G can be gracefully
labeled by labeling the vertices from the set {0, 1,
2, ... , q}, then all nontrivial trees are graceful.

PROOF: (Uses Induction on q)
Base Case: q = 1
0 1 1
Proof
Induction Hypothesis: Assume every nontrivial tree with q edges is
graceful.
Now look at tree G with q + 1 edges.
G is a tree, so has a vertex of degree 1, call it v.
Now look at G – v.
v only has degree 1, so deleting v is only removing one edge from G,
call it edge e.
So G – v has q edges.
A f d 1 b i G i d
A vertex of degree 1 cannot be a cut-vertex, so since G is connected
(it is a tree), G – v is connected.

Proof
G has no cycles (since it is a tree), so G
– v has no cycles.
So, G – v is a tree with q edges.
So by our induction hypothesis, G – v is graceful.
So the vertices of G – v can be labeled gracefully from the set {0, 1,
2, ... , q}, with the edges of G – v having values 1, 2, ... , q.
Now look again at G. Keep all the vertices (except v) labeled as
they were in the graceful labeling of G – v.
Thus the edges of G (except edge e) have values 1, 2, ... , q.
We know edge e is incident id tto v, so let uv be edge e.

Proof
u is already labeled some integer from the set {0, 1, 2, ... , q}, call
the integer u is labeled k.
Label vertex v with k + q + 1.
This is legal since all the other vertices of G are labeled from the
set {0, 1, 2, ... , q} and k + q + 1 > q, so no other vertex has this
label.
Then edge e has value | (k + q + 1) – k|= |q+1|= | q+1.
Therefore, the edges of G have the values 1, 2, ... , q, q + 1.
So the vertices of G are labeled ed with distinct integers, and the edges
have values 1, 2, ... , q + 1.
Thus, G is graceful.

Theorem 4
Theorem 4: If the fact that a (p,q) graph G can be
gracefully labeled by labeling the vertices from
the set of integers implies that G can be gracefully
labeled by labeling the vertices from the set {0, 1,
2, ... , q}, then all nontrivial trees are graceful.

Anyone Interested???
Kari.F.Lock@williams.edu

References
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Digraphs. Wadsworth: Belmont, CA. 1979. pg 51.
Chartrand, Gary & Lesniak, Linda. Graphs & Digraphs; second edition.
Wadsworth, Inc.: Belmont, CA. 1986. pgs 76-77.
Chartrand, G. & Lesniak, L. Graphs & Digraphs; third edition. Chapman
& Hall: London, UK. 1996. pgs 281-301.
Kevin Gong. http://kevingong.com/Math/GracefulGraphs.html. 10/30/02.
Weisstein, Eric W.
http://hades.ph.tn.tudelft.nl/Internal/PHServices/Documentation/M
athWorld/math/math/g/g226.htm. 10/30/02.
West, Douglas B. Introduction to Graph Theory. Prentice Hall: Upper
Saddle River, NJ. 1996. pgs 69-73.
West, Douglas B. Introduction to Graph Theory; 2 nd edition. Prentice Hall:
Upper Saddle River, NJ. 2001. pgs 89-94.