Two stars are better than one
Fred Cline, MWSC student
Jeffrey L. Poet, faculty project mentor
Missouri MAA, St. Joseph
April 1, 2005
In 2002, Sin-Min Lee and Man Kong 1 conjectured that every constellation graph
with an odd number of stars is super edge-magic. Constellations with three stars were
investigated by a group of MWSC student researchers. 2 This paper represents progress
toward classifying which constellation graphs with two stars are super edge-magic.
We begin with some definitions. First, an n-star is a graph on n  1 vertices such
that one vertex, called the central vertex, is connected to each of the other vertices with a
single edge and these are the only edges in the graph. Examples of a 3-star and a 4-star
are given below.
Second, a constellation is a union of star graphs. We denote the constellation
with two stars, an n-star and an m-star where n  m , by the ordered pair (n, m) . As an
example the constellation (3, 4) is given below.
Third, a graph with v vertices and e edges is called super edge-magic if the
vertices can be labeled with the integers 1, 2,
integers v  1, v  2,
,v and the edges can be labeled with the
, v  e such that the sum of the labels on any edge and its two
endpoints is the same regardless of which edge is chosen. The example below shows a
super edge-magic labeling of the constellation  3,4 . Note that for any edge, the sum of
the label on that edge and the labels on its two endpoints is 21.
Lee and Kong 1 showed that for any positive integers n and k, the constellation
 n, k (n  1)  is super edge-magic.
We provide evidence here to support our conjecture
that these are the only constellations with two stars that are super edge-magic. We do not
yet have a complete proof, but hope to soon.
To simplify our presentation, we next introduce the idea of a vertex-sum. For
each edge in a labeled graph, we define the vertex-sum of the edge to be the sum of the
labels of the two endpoints of the edge. Observe that a graph is super edge-magic if and
only if the set of vertex-sums of the graph is a set of consecutive integers. For the labeled
constellation (3,4) above, the set of vertex-sums is 6,8,10,5,7,9,11 .
The purpose of this paper is to show that constellations of the form
 4N  3,4M 1 are not super edge-magic for any positive integer values of N and M.
To do so, we assume an appropriate labeling of the vertices exists and then derive a
contradiction. To this end, suppose that a super edge-magic labeling of  4 N  3,4M  1
exists and that the central vertex of the  4 N  3 -star is labeled k and the central vertex of
the  4M  1 -star is labeled
. Then the total of the vertex-sums of the graph is
Total of vertex-sums  1  2 
  4 N  3  4 M  1  2     4 N  4  k   4 M  2  .
where the first part of the sum is the sum of each of the vertex labels one time, the second
part accounts for the use of k as an endpoint of 4N  3 edges (but one use has been
included in the first part of the sum), and the last part accounts for the multiple use of the
label
in a similar way. Note that the first part of the above computation is the sum of
the first 4N  4M  2 positive integers. We now use a familiar formula to rewrite this
sum and we consider the parity of our sum.
Total of vertex-sums 
 4 N  4M  2  4 N  4M  1 
2
 4 N  4  k   4M  2 
  2 N  2M  1 4 N  4M  1   4 N  4  k   4M  2 
.
 odd  even  even
 odd
We next work to derive a contradiction. Since the graph is assumed to be super
edge-magic and the given labeling is assumed to illustrate this fact, we know that the set
of  4 N  3   4M 1  4  N  M  1 vertex-sums form a set of consecutive integers.
Observe that the sum of any four consecutive integers is even, as illustrated by
p   p  1   p  2   p  3  4 p  6 . It follows that the sum of any set of consecutive
integers with some multiple of 4 elements must be even. This contradicts that the total of
the vertex-sums is odd. Therefore, any constellation of the form  4 N  3,4M  1 is not
super edge-magic. Note that the argument also proves that constellations of the form
 4M 1,4 N  3 are not super edge-magic because the fact that
4N  3  4M 1 was
never used in the argument.
Why are constellations with two stars better than those with one star? It is trivial
that a single n-star is super edge-magic as shown below.
On the other hand, some constellations with two stars are super edge-magic and others
are not, making constellations with two stars far more interesting.
(1) S.-M. Lee & M. Kong, On super-edge magic n-stars, Journal of Combinatorial
Mathematics and Combinatorial Computing, 40(2002), 87-96.
(2) J. Poet, V. Onkoba, D. Daffron, H. Goforth, & C. Thomas, On super-edge magic
labelings of unions of star graphs, Journal of Combinatorial Mathematics and
Combinatorial Computing, 53(2005), 49-63.