Unipancyclic Matroids

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

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)

Examples:

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 Markström’s paper, “A Note on Uniquely Pancyclic Graphs,” is an excellent source of information on this problem. • This problem appears in Bondy and Murty’s book Graph Theory with Applications.

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.

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.

e1 e2 e3 er-1er f1 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.

e2 e1 f1 e2 e1 f2 f1 e3 e4 Example: For the triangle, we have Example: For the second graph, we have

Example: The Octagons: Coming soon to a blueboard near you! e1 e2 f2 e7 f1 f3 e6 e3 e5 e4

Example: A 14-gon. e2 e1 e3 e13 f2 e4 e12 f3 f8 e5 e11 e6 e10 e9 f1 e7 e8

Now consider: • We don’t know r. • We don’t know k (the number of 1’s). • We don’t know where the 1’s of fk should overlap the other 1’s. • We don’t know whether these four f’s have an appropriate fk. • We don’t even know whether there are any such matroids!

On the other hand: • We doknow 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.

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.

Let’s try

Try this one:

We find:

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.)

This is not a tenable method for finding more! This is where MAPLE comes into play. There are two parts to the code: • The first part creates a loop that cycles through candidates for fk. • The second part tests whether the matrix with that fk represents a UPC matroid. MAPLE

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

Unipancyclic Matroids