Unipancyclic Matroids

1
Unipancyclic Matroids
PALS of Graph Theory and Combinatorics
(very) Preliminary Report

• Colin Starr, Mathematics
• Willamette University
• Joint work with Dr. Galen Turner, Louisiana Tech
• Tuesday, November 9, 2004

2
Definition A connected graph G on n vertices is
pancyclic if it has a cycle each size 3 through
n. G is uniquely pancyclic or unipancyclic (UPC)
if it has exactly one cycle of each size 3
through n.
Question For which n is there a UPC graph with n
vertices? (Entringer, 1973)
3
Examples
4
Notes

• A UPC graph on n vertices has n 2 cycles of
  sizes from 3 to n.

• UPC graphs are necessarily hamiltonian.

• It is very easy to find a UPC graph for any n if
  the connected requirement is dropped.

• Klas Markströms paper, A Note on Uniquely
  Pancyclic Graphs, is an excellent source of
  information on this problem.

• This problem appears in Bondy and Murtys book
  Graph Theory with Applications.

5
Theorem (Markström, et al) For n 56, these
are the only UPC graphs. For cycles of any size
plus at most five chords, these are the only UPC
graphs.
Joshua Hughes, a graduate student at Louisiana
Tech, is working on this problem for his
dissertation.
6
Definition A matroid M of rank r is unipancyclic
(UPC) if it has exactly one circuit of each size
3 through r 1.
We have been examining binary matroids as a
starting point.
7
The plan By analogy with the graphic case, we
begin with a hamiltonian circuit that is, a
circuit of size r 1. All of the edges but one
we label with ei, the standard basis vectors, and
the last as f1. We represent this with the
matrix below.
We then begin adding chords as appropriate.
8
Example For the triangle, we have
Example For the second graph, we have
9
Example The Octagons Coming soon to a blueboard
near you!
e1
e2
f2
e7
f1
f3
e3
e6
e4
e5
10
Example A 14-gon.
e1
e2
e3
e13
f2
e4
e12
f3
f8
e5
e11
e6
e10
e9
f1
e7
e8
11
Now consider

• We dont know r.

• We dont know k (the number of 1s).

• We dont know where the 1s of fk should overlap
  the other 1s.

• We dont know whether these four fs have an
  appropriate fk.

• We dont even know whether there are any such
  matroids!

12
On the other hand

• We do know k is not 2, 3, or 8.

• Since there are only 25 32 combinations of f1,
  f2, f3, f8, and fk and every circuit involves at
  least one of these, there are at most 31 circuits.

• Since the number of circuits is one less than the
  rank, the rank is at most 32.

13
Notice that f2 and f3 do not form a circuit such
a circuit would necessarily also contain e1
through e5. However, these are not minimally
dependent since e1, e2, and f2 form a circuit.
In fact, to determine whether a collection C of
f-columns corresponds to a circuit, we must test
every proper subset of C for dependence.
This is not fun.
14
Lets try
15
Try this one
16
We find
17
What went into the guessing?

• At first, not much!

• However, once we settle on a particular fk to
  try, we can compare combinations of the f-vectors
  to count circuits. From that we can deduce the
  rank. (32 is big, but not that big!)

• From this, we determine a system equations for
  which we seek a solution. (See overhead.)

18
This is not a tenable method for finding more!
This is where MAPLE comes into play.
There are two parts to the code

1. The first part creates a loop that cycles
   through candidates for fk.
2. The second part tests whether the matrix with
   that fk represents a UPC matroid.

MAPLE
19
The MAPLE code is available at
http//www.willamette.edu/cstarr/research.html
Thanks!