On The De Bruijn Torus Problem

A (kn;n)k-de Bruijn Cycle is a cyclic k-ary sequence with the property that every k-ary n-tuple appears exactly once contiguously on the cycle. A (kr, ks; m, n)k-de Bruijn Torus is a k-ary krXks toroidal array with the property that every k-ary m x n matrix appears exactly once contiguously on the torus. As is the case with de Bruijn cycles, the 2-dimensional version has many interesting applications, from coding and communications to pseudo-random arrays, spectral imaging, and robot self-location. J.C. Cock proved the existence of such tori for all m, n, and k, and Chung, Diaconis, and Graham asked if it were possible that r = s and m -= n for n even. Fan, Fan, Ma and Siu showed this was possible for k - 2. Combining new techniques with old, we prove the result for k \u3e 2 and show that actually much more is possible. The cases in 3 or more dimensions remain

A Classification of Tournaments Having an Acyclic Tournament as a Minimum Feedback Arc Set

Given a tournament with an acyclic tournament as a feedback arc set we give necessary and sufficient conditions for this feedback arc set to have minimum size

A Pebbling Game on Powers of Paths

Two Player Graph Pebbling is an extension of graph pebbling. Players Mover and Defender use pebbling moves, the act of removing two pebbles from one vertex and placing one pebble on an adjacent vertex, to win. If a specified vertex has a pebble on it, then Mover wins. If a specified vertex is pebble-free and there are no more valid pebbling moves, then Defender wins. The Two-Player Pebbling Number of a graph G, η(G), is the minimum m such that for every arrangement of m pebbles and for any specified vertex, Mover can win. We specify the winning player for powers of a path

On Two-Player Pebbling

Graph pebbling can be extended to a two-player game on a graph G, called Two-Player Graph Pebbling, with players Mover and Defender. The players each use pebbling moves, the act of removing two pebbles from one vertex and placing one of the pebbles on an adjacent vertex, to win. Mover wins if they can place a pebble on a specified vertex. Defender wins if the specified vertex is pebble-free and there are no more pebbling moves on the vertices of G. The Two-Player Pebbling Number of a graph G, η(G), is the minimum m such that for every arrangement of m pebbles and for any specified vertex, Mover can win. We specify the winning player for paths, cycles, and the join of certain graphs

The reversing number of a diagraph

AbstractA minimum reversing set of a diagraph is a smallest sized set of arcs which when reversed makes the diagraph acyclic. We investigate a related issue: Given an acyclic diagraph D, what is the size of a smallest tournament T which has the arc set of D as a minimun reversing set? We show that such a T always exists and define the reversing number of an acyclic diagraph to be the number of vertices in T minus the number of vertices in D. We also derive bounds and exact values of the reversing number for certain classes of acyclic diagraphs

Sum List Coloring 2 n Arrays

A graph is f-choosable if for every collection of lists with list sizes speci ed by f there is a proper coloring using colors from the lists. The sum choice number is the minimum over all choosable functions f of the sum of the sizes in f . We show that the sum choice number of a 2 n array (equivalent to list edge coloring K 2;n and to list vertex coloring the cartesian product K 2 2K n ) is n + d5n=3e