Kari Lock. Hudson River Undergraduate Mathematics Conference 2003

Transcription

1 Kari Lock Williams College Hudson River Undergraduate R g Mathematics Conference 2003

2 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

3 Example: K

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

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

6 Star Graphs Theorem: Every star graph his graceful.

7 Path Graphs Theorem: Every path graph his graceful.

8 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.

9 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.

10 Path Graphs Theorem: Every path graph his graceful.

11 0 Cycle Graphs 2 3 => NOT GRACEFUL Theorem:C p is graceful if and only if 4 p or 4 (p+1)

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

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

14 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...

15 More Graceful Graphs Trees???

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

17 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???

18 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!

19 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.

20 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.

21 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.

22 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.

23 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.

24 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!

25 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.

26 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.

27 PROOF: (Uses Induction on q) Base Case: q = 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.

28 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.

29 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.

30 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.

31 Anyone Interested???

32 References Behzad, Mehdi, Chartrand, Gary, & Lesniak-Foster, Linda. Graphs & Digraphs. Wadsworth: Belmont, CA pg 51. Chartrand, Gary & Lesniak, Linda. Graphs & Digraphs; second edition. Wadsworth, Inc.: Belmont, CA pgs Chartrand, G. & Lesniak, L. Graphs & Digraphs; third edition. Chapman & Hall: London, UK pgs Kevin Gong. 10/30/02. Weisstein, Eric W. athworld/math/math/g/g226.htm. 10/30/02. West, Douglas B. Introduction to Graph Theory. Prentice Hall: Upper Saddle River, NJ pgs West, Douglas B. Introduction to Graph Theory; 2 nd edition. Prentice Hall: Upper Saddle River, NJ pgs