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Unipancyclic Matroids Colin Starr, Mathematics Willamette University Joint work with Dr. Galen Turner, Louisiana Tech Tuesday, November 9, 2004 PALS of Graph Theory and Combinatorics (very) Preliminary Report
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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)
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Examples:
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Notes : A UPC graph on n vertices has n – 2 cycles of sizes from 3 to n. This problem appears in Bondy and Murty’s book Graph Theory with Applications. Klas Markström’s paper, “A Note on Uniquely Pancyclic Graphs,” is an excellent source of information on this problem. It is very easy to find a UPC graph for any n if the connected requirement is dropped. UPC graphs are necessarily hamiltonian.
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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.
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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.
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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 e i, the standard basis vectors, and the last as f 1. We represent this with the matrix below. e 1 e 2 e 3 e r-1 e r f 1 We then begin adding “chords” as appropriate.
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Example: For the triangle, we have Example: For the second graph, we have e1e1 e2e2 f1f1 e1e1 e2e2 f1f1 e3e3 e4e4 f2f2
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Example: The Octagons: Coming soon to a blueboard near you! e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 f1f1 f2f2 f3f3
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Example: A 14-gon. e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 e9e9 e 10 e 11 e 12 e 13 f1f1 f2f2 f3f3 f8f8
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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 f k should overlap the other 1’s. We don’t know whether these four f’s have an appropriate f k. We don’t even know whether there are any such matroids!
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We do know k is not 2, 3, or 8. Since there are only 2 5 = 32 combinations of f 1, f 2, f 3, f 8, and f k 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. On the other hand:
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Notice that f 2 and f 3 do not form a circuit: such a circuit would necessarily also contain e 1 through e 5. However, these are not minimally dependent since e 1, e 2, and f 2 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.
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Let’s try
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Try this one:
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We find:
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What went into the guessing? At first, not much! However, once we settle on a particular f k 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.)
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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 f k. 2.The second part tests whether the matrix with that f k represents a UPC matroid. MAPLE
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The MAPLE code is available at http://www.willamette.edu/~cstarr/research.html Thanks!
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