Infinite Hamiltonian paths in Cayley diagraphs of hyperbolic symmetry groups

AbstractThe hyperbolic symmetry groups [p,q], [p,q]+, and [p+, q] have certain natural generating sets. We determine whether or not the corresponding Cayley digraphs have one-way infinite or two-way infinite directed Hamiltonian paths. In addition, the analogous Cayley graphs are shown to have both one-way infinite and two-way infinite Hamiltonian paths

Harmonious groups

In this paper we introduce a method of sequencing the elements of a finite group that gives rise to a complete mapping of the group. Our definition was motivated by the concept of a harmonious graph invented by Graham and Sloane. Our concept has several connections to graph theory and as an application we complete the characterization of elegant cycles begun by Chang, Hsu, and Rogers. Our definitions are also variations of the notion of an R-sequenceable group first introduced by Ringel in his solution of the map coloring problem for all compact 2-dimensional manifolds except the sphere and expanded upon by Friedlander, Gordon, and Miller.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/29431/1/0000512.pd

Buchsbaum and Eulerian complexes

AbstractLet Δ be a finite simplicial complex, and K [Δ] its Stanley-Reisner ring. We show that if Δ is Eulerian, then every depth of K[Δ] is possible, and we find all possible depths of K [Δ] for Buchsbaum Eulerian Δ, dealing with the orientable and nonorientable cases separately. In addition, we find all possible betti sequences of Buchsbaum-Eulerian, Eulerian, and semi-Eulerian complexes

Almost All Complete Binary Prefix Codes Have a Self-Synchronizing String

The probability that a complete binary prefix code has a selfsynchronizing string approaches one, as the number of codewords tends to infinity

Self-Synchronization of Huffman Codes

Variable length binary codes have been frequently used for communications since Huffman’s important paper on constructing minimum average length codes. One drawback of variable length codes is the potential loss of synchronization in the presence of channel errors. However, many variable length codes seem to possess a “self-synchronization” property that lets them recover from bit errors. In particular, for some variable length codes there exists a certain binary string (not necessarily a codeword) which automatically resynchronizes the code. That is, if a transmitted sequence of bits is corrupted by one or more bit errors, then as soon as the receiver by random chance correctly detects a self-synchronizing string, the receiver can continue properly parsing the bit sequence into codewords. Most commonly used binary prefix codes, including Huffman codes, are “complete”, in the sense that the vertices in their decoding trees are either leaves or have two children. An open question has bee