Kari LockW ll llWilliams College

Hudson River Undergraduate h f

R gMathematics Conference 2003

D fi i iDefinitionDefinition: A graceful labeling is aDefinition: A graceful labeling is a labeling of the vertices of a graph with distinct integers from the set {0 1 2distinct integers from the set {0, 1, 2, ... , q} (where q represents the number of edges) such that...g )

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

E l KExample: K300

2 3

2 31

D fi i iDefinition

Definition: A graph G is graceful if and only if...

G can be labeled gracefully.

Are The Following Graphs Graceful?Are The Following Graphs Graceful?• Star Graphs?

• Path Graphs?

C l G h ?•Cycle Graphs?

• Complete Graphs?

• Complete Bipartite Graphs?

• Wheel Graphs?• Wheel Graphs?

• Polyhedral Graphs?

• Trees???

S G hStar Graphs

1 21 2

7 3

0

1 2

7 3

6 40

6 4

6 4

5

5

Th E h i f lTheorem: Every star graph is graceful.

Path Graphs

Th E h h i f lTheorem: Every path graph is graceful.

Path GraphsPath GraphsProof: Let G be a path graph.

• 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.

Th + 1 ti th fi t t ill l b l it’• There are q + 1 vertices, so the first set will label 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 GraphsPath Graphs• With the vertices labeled in this manner the edges• 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•Thus, this is a graceful labeling, so G is graceful.

•Therefore, all path graphs are graceful.

Path GraphsPath Graphs0

3

3

3

1

2

1

1

2

Th E h h i f lTheorem: Every path graph is graceful.

Cycle Graphs0

2 3 => NOT GRACEFUL

2 3

2 3

1

=> NOT GRACEFUL

0 33

4 2

4

2

1

Theorem: C is graceful if and only if 4|p or 4|(p+1)

4 22

Theorem: Cp is graceful if and only if 4|p or 4|(p+1)

E l i G hEulerian Graphs

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

Complete GraphsComplete Graphs20

2 30 1 1

0 2

6

2

34

2 31 6 515

Theorem: K2, K3, K4 are the only graceful complete graphs.

More Graceful GraphsComplete Bipartite Graphs

Wheel Graphs

Polyhedral Graphs y p

Peterson Graph

All graphs of order 4 or less

All graphs of order 5 except...

More Graceful GraphsMore Graceful GraphsTrees???

T E lTree Example

Def: A tree is a connected graph with no cyclesg p y

TreesTreesKotzig’s Conjecture: Every nontrivial tree is gracefulKotzig 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

ll t b t it!all trees, but no one can prove it!

=> BIG QUESTION FOR GRACEFUL=> BIG QUESTION FOR GRACEFUL GRAPHS: IS EVERY TREE GRACEFUL???

Definition of Graceful???Definition of Graceful???Def: A graceful labeling is a labeling of the vertices of a graph Def: A graceful labeling is a labeling of the vertices of a graph g g g g pwith 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

g g g g pwith 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 qvalue | f(u) – f(v) |, the edges are labeled 1, 2, ... , q

• integers from the set {0, 1, 2, ... , q}

value | f(u) – f(v) |, the edges are labeled 1, 2, ... , q

• integers

• nonnegative integers

• positive integers ???OH NO!Maybe they are all the same!!!

C j 1Conjecture 1Conjecture 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 integersintegers.

Conjecture 1Proof: 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) |.

h h d l d hi i ill f l l b liThus, the edge values are preserved so this is still a graceful labeling.

Th 1Theorem 1

Theorem 1: If a graph G can be gracefully g p g ylabeled 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.

C j 2Conjecture 2

Conjecture 2: If a graph G can be gracefully labeled by labeling the vertices from the set oflabeled 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.

Th 2Theorem 2

Theorem 2: If a graph G can be gracefully i i f flabeled by labeling the vertices from the set of

integers, then G can be gracefully labeled by labeling the vertices from the set of positivelabeling the vertices from the set of positive

integers.

D fi i i f G f l???Definition 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 qgiven the value | u-v |, the edges are labeled 0, 1, 2, ... , q (where q is the number of edges).

• integersintegers

• nonnegative integers

iti i t

INTERCHANGEABLE

IN THE DEFINITION!• positive integers

• integers from the set {0, 1, 2, ... , q}

C j 3Conjecture 3Conjecture 3: If a (p,q) graph G can be gracefully labeled by labeling the vertices from g y y gthe set of integers, then G can be gracefully labeled by labeling the vertices from the set {0 1 2 }{0, 1, 2, ... , q}.

Unfortunately, this is still a conjecture.

Importance of Conjecture 3Importance of Conjecture 3If Conjecture 3 is true, I will be able to proveIf 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 b f ll l b l d b l b li th ti fbe 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,labeled by labeling the vertices from the set {0, 1, 2, ... , q}, then all nontrivial trees are graceful.

Prooff

0 1 1PROOF: (Uses Induction on q)Base Case: q = 1Induction Hypothesis: Assume every nontrivial tree with q edges is graceful.

Base Case: q = 1

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 ecall it edge e.

So G – v has q edges.

A f d 1 b i G i dA vertex of degree 1 cannot be a cut-vertex, so since G is connected (it is a tree), G – v is connected.

Prooff

G has no cycles (since it is a tree) so G – v has no cyclesG has no cycles (since it is a tree), so G v has no cycles.

So, G – v is a tree with q edges.

S b i d ti h th i G i f lSo 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 q2, ... , 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 – vthey were in the graceful labeling of G v.

Thus the edges of G (except edge e) have values 1, 2, ... , q.

W k d i i id t t l t b dWe know edge e is incident to v, so let uv be edge e.

Prooffu is already labeled some integer from the set {0, 1, 2, ... , q}, call

the integer u is labeled k. gLabel vertex v with k + q + 1.This is legal since all the other vertices of G are labeled from the gset {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.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 with distinct integers, and the edges So e ve ces o G a e abe ed w d s c ege s, a d e edgeshave values 1, 2, ... , q + 1.Thus, G is graceful.

Th 4Theorem 4

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

A I d???Anyone Interested???

Kari.F.Lock@williams.edu

ReferencesBehzad, Mehdi, Chartrand, Gary, & Lesniak-Foster, Linda. Graphs &

Digraphs. Wadsworth: Belmont, CA. 1979. pg 51.

Chartrand Gary & Lesniak Linda Graphs & Digraphs; second editionChartrand, Gary & Lesniak, Linda. Graphs & Digraphs; second edition. Wadsworth, Inc.: Belmont, CA. 1986. pgs 76-77.

Chartrand, G. & Lesniak, L. Graphs & Digraphs; third edition. Chapman , , p g p ; p& 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/MathWorld/math/math/g/g226.htm. 10/30/02.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; 2nd edition. Prentice Hall: Upper Saddle River, NJ. 2001. pgs 89-94.