Perfect codes, NP-completeness, and Towers of Hanoi graphs

The set of codewords for a standard error-correcting code can be viewed as subset of the vertices of a hypercube. Two vertices are adjacent in a hypercube exactly when their Hamming distance is 1. A code is a perfect-error-correcting code if no two codewords are adjacent and every non-codeword is adjacent to exactly one codeword. Since such a code can be described using only vertices and adjacency, the de finition applies to general graphs rather than only to hypercubes. How does one decide if a graph can support a perfect 1-error-correcting code? The obvious way to show that such a code exists is to display the code. On the other hand, it seems difficult to show that a graph does not support such a code. We show that this intuition is correct by showing that to determine if a graph has a perfect 1-error-correcting code is an NP-complete problem. The proof is by reduction from 3-SAT. To show that perfect codes in graphs is not vacuous, we give an in nite family of graphs so that each graph in the family has a perfect 1-error-correcting code. Our graphs are based on the Towers of Hanoi puzzle, so that, each vertex is a confi guration of the puzzle and two vertices are adjacent when they are one legal move apart. We give a recursive construction which determines which vertices are codewords. There is a natural correspondence between the hypercube vertices and the binary strings, and there is a natural correspondence between Tower of Hanoi con guration and ternary strings. Our recursive construction also speci es which ternary strings are codewords. We characterize the codewords as the set of ternary strings with an even number of 1's and an even number of 2's. As part of this characterization, we show that there is essentially one perfect 1-error-correcting code for each n. There is a unique code when n is even, but the code is only unique up to a permutation of 0, 1, and 2 when n is odd. We show that error-correction can be accomplished by a nite state machine which passes over the ternary string twice, and that this machine is xed independent of the length of the string. Encoding and decoding are the mappings between integers and codewords, and vice-versa. While algorithms for such mappings can be derived directly from the recursive construction, we show that encoding/decoding can be carried out by multiplication/division by 4 and error-correction. So error-correction, encoding, and decoding can all be done in time θ(n) for code strings of length n in these codes

NCUWM Talk Abstracts 2010

Dr. Bryna Kra, Northwestern University “From Ramsey Theory to Dynamical Systems and Back” Dr. Karen Vogtmann, Cornell University “Ping-Pong in Outer Space” Lindsay Baun, College of St. Benedict Danica Belanus, University of North Dakota Hayley Belli, University of Oregon Tiffany Bradford, Saint Francis University Kathryn Bryant, Northern Arizona University Laura Buggy, College of St. Benedict Katharina Carella, Ithaca College Kathleen Carroll, Wheaton College Elizabeth Collins-Wildman, Carleton College Rebecca Dorff, Brigham Young University Melisa Emory, University of Nebraska at Omaha Avis Foster, George Mason University Xiaojing Fu, Clarkson University Jennifer Garbett, Kenyon College Nicki Gaswick, University of Nebraska-Lincoln Rita Gnizak, Fort Hays State University Kailee Gray, University of South Dakota Samantha Hilker, Sam Houston State University Ruthi Hortsch, University of Michigan Jennifer Iglesias, Harvey Mudd College Laura Janssen, University of Nebraska-Lincoln Laney Kuenzel, Stanford University Ellen Le, Pomona College Thu Le, University of the South Shauna Leonard, Arkansas State University Tova Lindberg, Bethany Lutheran College Lisa Moats, Concordia College Kaitlyn McConville, Westminster College Jillian Neeley, Ithaca College Marlene Ouayoro, George Mason University Kelsey Quarton, Bradley University Brooke Quisenberry, Hope College Hannah Ross, Kenyon College Karla Schommer, College of St. Benedict Rebecca Scofield, University of Iowa April Scudere, Westminster College Natalie Sheils, Seattle University Kaitlin Speer, Baylor University Meredith Stevenson, Murray State University Kiri Sunde, University of North Carolina Kaylee Sutton, John Carroll University Frances Tirado, University of Florida Anna Tracy, University of the South Kelsey Uherka, Morningside College Danielle Wheeler, Coe College Lindsay Willett, Grove City College Heather Williamson, Rice University Chengcheng Yang, Rice University Jie Zeng, Michigan Technological Universit

On perfect codes in Cartesian products of graphs

AbstractAssuming the existence of a partition in perfect codes of the vertex set of a finite or infinite bipartite graph G we give the construction of a perfect code in the Cartesian product G□G□P2. Such a partition is easily obtained in the case of perfect codes in Abelian Cayley graphs and we give some examples of applications of this result and its generalizations

Covering codes in Sierpinski graphs

Graphs and AlgorithmsInternational audienceFor a graph G and integers a and b, an (a, b)-code of G is a set C of vertices such that any vertex from C has exactly a neighbors in C and any vertex not in C has exactly b neighbors in C. In this paper we classify integers a and b for which there exist (a, b)-codes in Sierpinski graphs

Perfect codes in direct products of cycles—a complete characterization

AbstractLet be a direct product of cycles. It is known that for any r⩾1, and any n⩾2, each connected component of G contains a so-called canonical r-perfect code provided that each ℓi is a multiple of rn+(r+1)n. Here we prove that up to a reasonably defined equivalence, these are the only perfect codes that exist

The Tutte polynomial of the Sierpinski and Hanoi graphs

We study the Tutte polynomial of two infinite families of finite graphs: the Sierpi\'{n}ski graphs, which are finite approximations of the well-known Sierpi\'{n}ski gasket, and the Schreier graphs of the Hanoi Towers group $H^{(3)}$ acting on the rooted ternary tree. For both of them, we recursively describe the Tutte polynomial and we compute several special evaluations of it, giving interesting results about the combinatorial structure of these graphs.Comment: 30 pages; title changed; revised exposition in the second version but results unchanged. arXiv admin note: substantial text overlap with arXiv:1010.290