Download golomb rulers and graceful graphs

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.
• OGR- (optimal Golomb ruler) is a Golomb ruler that
is of minimum length for a given number of marks.
Notice that not each
Golomb ruler is an
OGR
• 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.
The length of the ruler is equal to the sum of the
first-order distances.
There are  n2  possible distances measured between
marks.
The largest distance measured by the ruler is  n2 .
n( n  1)
i
 n

2 .
We know that
2
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{1,2,..., n  1} .
We now place the first-order distances in the ruler
beginning with the distance 1.
n 1
i 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 firstorder 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.
The distance 2 cannot be adjacent to the distance n2.
Thus the distance 2 must be adjacent to the n - 1
distance on the other side of the ruler from the
size 1 distance.
n 2
So there is a mark at  2  .
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 n1  3 we get a contradiction. Thus n  4 .
• 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 a1, a2 ,..., an to the marks of the ruler
from left to right.
Formally, the aij entry of the triangle is | a  a | .
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 n  n matrix
where the element aij is (a  a ) mod k .
j i
i
j
j
• 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
1
10
3
12
6
4
2
9
8
10
11
12
17
5
7
13
16
17
0
1
4
10
12
17
1
0
3
9
11
16
4
3
0
6
8
13
10
9
6
0
2
7
12
11
8
2
0
5
17
16
13
7
5
0
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 f from the vertices of a
graph G to the set {0,1,…,|E(G)|} such that the
edge label function g 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 K 2 n1 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: K 2 n1 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 3.4:For all positive integer a and b, the
complete bipartite graph K a ,b 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.
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 e 2 1 .
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 K n is graceful if and
only if there is a perfect Golomb ruler with n
marks .
Proof: suppose there is a graceful labeling of K n
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 K n .
For each edge in K n there is a corresponding
distance between marks in R.
Since the values of the edges of K n 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. Let G be the complete graph on
n vertices. Let f assign the positions of the
marks in R bijectively to the vertices of K n as
the value of the vertex in G.
Give each edge uv in K n 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.