Euler Circuits, Dna Sequencing by Hybridization, and a New Graph Polynomial That Counts Euler Circuit Decompositions

I will survey in this talk two papers written in collaboration with Bela Bollobas, Don Coppersmith, and Greg Sorkin. The second of these papers studies how Euler circuits for a type of Eule-rian, directed graphs, can be counted by a recursion relation. Generalizing this recursion relation deenes a polynomial, the interlace polynomial, on any undirected graph. We introduce this new polynomial and explore some of its properties. In the rst paper surveyed in my talk, I present some new results related to the sequencing by hybridization, a method of reconstructing a long DNA string |that is, guring out its nucleotide sequence| from knowledge of its short substrings. Unique reconstruction is not always possible, and the goal of this paper is to study the number of reconstructions of a random DNA string, under an appropriate probabilistic model. For a given string, the number of reconstructions is determined by the pattern of repeated substrings; in an appropriate limit substrings will occur at most twice, so the pattern of repeats is given by a pairing: a string of length 2n in which each symbol occurs twice. A pairing induces a 2-in,2-out graph, whose directed edges are deened by successive symbols of the pairing |for example the pairing ABBCAC induces the graph with edges AB, BB, BC, and so forth| and the number of reconstructions is simply the number of Euler circuits in this 2-in 2-out graph. The original problem is thus transformed into a question about pairings: how many n-symbol pairings have k Euler circuits? We show how to compute this function, in closed form, for any xed k, and we present the functions explicitly for k = 1 to 9. The key is a decomposition theorem: the Euler \circuit number" of a pairing is the product of the circuit numbers of \component" sub-pairings. These components come from connected components of the \interlace graph", which has the pairing's symbols as vertices, and edges when symbols are \interlaced". (A and B are interlaced if the pairing has the form We carry these results back to the original question about DNA strings, with a total variation distance upper bound 1