Kari Lock WllWilliams College Hudson RRgiver Undergraduate Mathfhematics Conference 2003 DfiiiDefinition Definition:A: A graceful 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 edgg)es) 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 ElExample: K3 0 2 3 2 1 3 DfiiiDefinition Definition: A graph G is graceful if and only if... G can be labeled gracefully. Are The Following Graphs Graceful? • Star Graphs? • Path Graphs? •ClGCycle Grap h?hs? • Complete Graphs? • Complete Bipartite Graphs? • Wheel Graphs? • Polyhedral Graphs? • Trees??? Star GhGraphs 1 2 121 2 7 7 3 3 646 0 4 6 5 4 5 Theorem: EhiflEvery star graph is graceful. Path Graphs Theorem: EhhiflEvery path graph is graceful. Path Graphs Proof: 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. • There are q + 1 verti ces, so th e fi rst set will l ab el it’ s vertices with numbers from the set • {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 0 3 3 2 1 1 2 Theorem: EhhiflEvery path graph is graceful. Cycle Graphs 0 2 3 => NOT GRACEFUL 231 033 4 1 4 2 2 Theorem:C: Cp is graceful if and only if 4|p or 4|(p+1) ElEuleri an Graph s Theorem: If G is a (p, q) graceful Eulerian graph, then 4|q or 4|(q+1) . Complete Graphs 0 022 0 1 1 4 23 6 3 5 23 1 651 Theorem: K2, K3, K4 are the only graceful complete graphs. More Graceful Graphs ¾ Complete Bipartite Graphs ¾ Wheel Graphs ¾ Polyyphedral Graphs ¾ Peterson Graph ¾ All graphs of order 4 or less ¾ All graphs of order 5 except... More Graceful Graphs ¾ Trees??? Tree Exampl e Def: A tree is a connected ggpraph with no c ycles 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 t rees, b ut no one can prove it! => BIG QUESTION FOR GRACEFUL GRAPHS: IS EVERY TREE GRACEFUL??? Definition of Graceful??? Def: A gggraceful labeling is a labelingggp of the vertices of a graph with distinct integers from ththee 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 • positive integers Maybe they areOH all??? NO! the same!!! CjConjecture 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, t he e dge va lues are preserve d so t his i s still a gracef ul l ab eli ng. Theorem 1 Theorem 1: If a ggpraph G can be gracefull 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. CjConjecture 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 labeliiffing the vertices from the set of integers, then G can be gracefully labeled by labeling the vertices from the set of positive integers. De fini ti on of Gracef ul??? 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 INTERCHANGEABLE • nonnegative integers IN THE DEFINITION! • positive i nt egers • integers from the set {0, 1, 2, ... , q} CjConjecture 3 Conjecture 3: If a (p,q) graph G can be ggyracefully labeled by yg 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 grace flllbldbfully labeled by lblilabeling th e ver tices 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 PROOF: (Uses Induction on q) Base Case: q = 1 0 1 1 9 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. Afd1bA vertex of degree 1 cannot be a cut-vertex, so siGiince G is connecte d (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. SbSo by our idtihinduction hypoth thiesis, G – v iflis 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. WkWe know e dge e iiis incid idttent to v, so l ltet uv b e ed ge 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|k | = |q+1|| q + 1 | = q+1.q + 1. Therefore, the edges of G have the values 1, 2, ... , q, q + 1. So the ve rtices o f G a re labe led w ith d is tinc t intege rs, a nd 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 gracefiiffully 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??? [email protected] References Behzad, Mehdi, Chartrand, Gary, & Lesniak-Foster, Linda. Graphs & Digraphs. Wadsworth: Belmont, CA. 1979. pg 51. Chartrand, Gary & Lesniak , Linda . Graphs & Digraphs; second edition. Wadsworth, Inc.: Belmont, CA. 1986. pgs 76-77. Chartrand,,,pgp; 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; 2nd edition. Prentice Hall: Upper Saddle River, NJ. 2001. pgs 89-94..