GOLOMB RULERS AND GRACEFUL GRAPHS BRIAN BEAVERS

CONTENT: Introduction Rulers Graph Labeling Connections

Introduction At first one may wonder how rulers and graphs are related. Rulers –measures distances between objects Graphs- gives us a sense of how things connect to each other.

Rulers: A ruler is a straightedge containing labeled marks and is used to measure distances. One end is the zero end, and the distance from the zero end to the other end is the length of the ruler. The ruler as marks at a specified distance interval along the length of the ruler.

Our goal: Suppose we take n-1 distinct marks placed at integer distances from the zero end of the ruler. We want to find a ruler such that the distance between any two marks is distinct. Notice that we treat the zero end as another mark so that we have n marks. This type of ruler is called Golomb ruler.

Notice that not each Golomb ruler is an OGR OGR- (optimal Golomb ruler) is a Golomb ruler that is of minimum length for a given number of marks. Perfect Golomb ruler- a ruler in which the set of distances between marks is every positive integer up to and including the length of the ruler.

Theorem 2.1: There is a perfect Golomb ruler on n marks if and only if n is 1,2,3 or 4. Proof: A ruler with 0 marks is a perfect Golomb ruler trivially. A ruler with marks at 0 and 1 is a perfect Golomb ruler. A ruler with marks at 0,1 and 3 is a perfect Golomb ruler. Finally, a ruler with marks at 0,1,4 and 6 is a perfect Golomb ruler.

Now we want to show that there are not perfect Golomb rulers with more then 4 marks… First we make a few definitions: 1-First-order distance- a distance between consecutive marks. 2-Kth-order distance- a distance between two marks that have k-1 marks between them. 3-We say that two distances are adjacent if they share exactly one mark.

Let R be a perfect Golomb ruler Let R be a perfect Golomb ruler. The length of the ruler is equal to the sum of the first-order distances. There are possible distances measured between marks. The largest distance measured by the ruler is . We know that . So the distances from 1 to n- 1 can fill the ruler exactly as first-order distances. Thus, the set of first order distances is . We now place the first-order distances in the ruler beginning with the distance 1.

The distance 1 must be adjacent only to the n-1 distance. So the distance 1 must be at an end of the ruler. Now, we must place the distance 2 as a first-order distance. The distance 2 cannot use one of the marks of the distance 1 (why?). The distance 2 cannot be adjacent to a distance less than n – 2.

Q.E.D The distance 2 cannot be adjacent to the distance n-2. Thus the distance 2 must be adjacent to the n - 1 distance on the other side of the ruler from the size 1 distance. So there is a mark at . Including any other first-order distance between ones already placed would yield a contradiction. Thus no other first-order distances exist. We now have that if we get a contradiction. Thus . Q.E.D

A difference triangle for a ruler is formed by arranging the distances between marks in the ruler as a triangular matrix of numbers. We give labels to the marks of the ruler from left to right. Formally, the entry of the triangle is . If every entry in the triangle is distinct, then the ruler is a Golomb ruler. For our set of marks and a positive integer k, The difference table mod k is a matrix where the element is .

A distinct difference set mod k is a set of integer such that every entry in its difference table is distinct (except for the main diagonal). We can use these distinct difference sets to generate Golomb rulers ( the upper triangle of the table is the difference triangle ).

0 1 4 10 12 17 1 3 6 2 5 4 9 8 7 10 11 13 12 16 17 17 12 10 4 1 16 11 9 3 13 8 6 7 2 5

Applications: Golomb rulers are used to generate self orthogonal codes. Reduce ambiguities in X-ray crystallography. Create unique labels for paths in communications networks. Among other applications.

Graph Labeling A labeling of a graph is an assignment of values to the vertices and edges of graphs. A β-valuation, or graceful labeling is an injective vertex label function from the vertices of a graph G to the set {0,1,…,|E(G)|} such that the edge label function defined by g(e)=|f(u)-f(v)| where e is an edge having endpoints u and v, is a bijection from V(G) to {1,2,…,E(G)}.

A simple graph that has a graceful labeling is called a graceful graph.

Conjecture 3.1: the complete graph can be cyclically decomposed into 2n+1 subgraphs isomorphic to a given tree with n edges. This conjecture implies the following theorem relating to graceful graphs: Theorem 3.2: can be cyclically decomposed into 2n+1 subgraphs isomorphic to a given tree T with n edges if T is graceful. This motivated the search for a proof that all trees are graceful.

Conjecture 3. 3: All trees are graceful. Theorem 3 Conjecture 3.3: All trees are graceful. Theorem 3.4:For all positive integer a and b, the complete bipartite graph is graceful. Proof: It suffices find a numbering. Consider the two sets of vertices A and B, containing a and b vertices, respectively. Assign the vertices in set A the numbers 0,1,..a-1 and assign the vertices in set B the numbers a,2a,3a,..,ba. In this way, every integer from 1 to ab has a unique representation as a difference between a number in B and a number in A.

Theorem 3. 5: All caterpillars are graceful Theorem 3.5: All caterpillars are graceful. Definition: a caterpillar is a tree such that if all the vertices of degree 1 (leaves) are removed, the resulting subgraph is a path.

Golomb also proved the following necessary conditions for a graph to be graceful: Theorem 3.6: Let G be a graceful graph with n vertices and e edges. Let the vertices be partitioned into two sets E and O having, respectively, the vertices with even and odd labels. Then the number of edges connecting vertices in E with vertices in O is exactly .

Theorem 3. 7: Let G be an Eulerian graph Theorem 3.7: Let G be an Eulerian graph. If |E(G)| is equivalent to 1 0r 2 modulo 4, then G does not have a graceful labeling.

Connections: We now demonstrate the relationship between Golomb Rulers and Graceful graphs. Theorem 4.1: The graph is graceful if and only if there is a perfect Golomb ruler with n marks . Proof: suppose there is a graceful labeling of Let |E(G)|=m .let f be the injection from V(G) to {0,1,…,m} induced by the graceful labeling.

Let R be a ruler with marks at f(v) for each vertex v in Let R be a ruler with marks at f(v) for each vertex v in . For each edge in there is a corresponding distance between marks in R. Since the values of the edges of take on every value of S ={1,2,…m} exactly once, each value of S is a distance in R exactly once. Therefore, R is a perfect Golomb ruler.

Conversely, suppose R is a perfect Golomb ruler with n marks Conversely, suppose R is a perfect Golomb ruler with n marks. Let G be the complete graph on n vertices. Let f assign the positions of the marks in R bijectively to the vertices of as the value of the vertex in G. Give each edge uv in the value |f(u)-f(v)| . Each distance in R gets mapped to an edge value in G. These values are taken on bijectively from S. Thus G has a graceful labeling.

Theorem 4.2: A clique in a graceful graph G induces a Golomb ruler with the same number of marks as the number of vertices in the clique. a clique in an undirected graph G, is a set of vertices V, such that for every two vertices in V, there exists an edge connecting the two. Theorem 4.1 and Theorem 2.1 imply that there are no graceful complete graphs on more than 4 vertices.

Summary.. This time, let us begin with a Golomb ruler and look at the corresponding labeled complete graph. If we add a few vertices and edges to make up for the missing distances, we can obtain a graceful graph that has the original complete graph as a clique. This larger graph induces the Golomb ruler we started with. To sum up, we can lift ruler problems to questions about graceful graphs by using the correspondence between rulers and labeled complete graphs. This correspondence gives us that Golomb rulers are equivalent to complete subgraphs of graceful graphs.