Garth Isaak

• Discrete Mathematics
• 1994
A graph is said to be representable modulo n if its vertices can be labelled with distinct integers between 0 and nâˆ’ 1 inclusive such that two vertices are adjacent if and only if their labels are relatively prime to n. The representation number of graph G is the smallest n representing G. We review known results and investigate representation numbers forâ€¦ (More)
• Discrete Applied Mathematics
• 1995
We answer a question of Golumbic, Monma and Trotter by constructing proper tolerance graphs that are not unit tolerance graphs. An infinite family of graphs that are minimal in this respect is specified.
A graph is f -choosable if for every collection of lists with list sizes specified by f there is a proper coloring using colors from the lists. We characterize f -choosable functions for block graphs (graphs in which each block is a clique, including trees and line graphs of trees). The sum choice number is the minimum over all choosable functions f of theâ€¦ (More)
The digraphs P (n, k) have vertices corresponding to length k permutations of an n set and arcs corresponding to (k + 1) permutations. Answering a question of Starling, Klerlein, Kier and Carr we show that these digraphs are Hamiltonian for k â‰¤ n âˆ’ 3. We do this using restricted Eulerian cycles and the fact that P (n, k) is nearly the line digraph of P (n,â€¦ (More)
• Discrete Mathematics
• 1998
We say any order ~ is a tolerance order on a set of vertices if we may assign to each vertex x an interval Ix of real numbers and a real number tx called a tolerance in such a way that x~,y if and only if the overlap of Ix and ly is less than the minimum of t~ and ty and the center of I~ is less than the center of Iy. An order is a bitolerance order if andâ€¦ (More)
• Discrete Mathematics
• 1999
We examine powers of Hamiltonian paths and cycles as well as Hamiltonian (power) completion problems in several highly structured graph classes. For threshold graphs we give efficient algorithms as well as sufficient and minimax toughness like conditions. For arborescent comparability graphs we have similar results but also show that for one type ofâ€¦ (More)
A graph is f -choosable if for every collection of lists with list sizes specified 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 Ã— n array (equivalent to list edge coloring K2,n and to list vertexâ€¦ (More)
We examine the size s(n) of a smallest tournament having the arcs of an acyclic tournament on n vertices as a minimum feedback arc set. Using an integer linear programming formulation we obtain lower bounds s(n) â‰¥ 3n âˆ’ 2 âˆ’ blog2 nc or s(n) â‰¥ 3n âˆ’ 1 âˆ’ blog2 nc, depending on the binary expansion of n. When n = 2k âˆ’ 2t we show that the bounds are tight withâ€¦ (More)
• Discrete Mathematics
• 2008
We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph Ks,t which admits only the identity automorphism. In particular this allows us to determine the distinguishing number of the Cartesian product of complete graphs. The distinguishing number of a graph is the minimum number of colors needed to labelâ€¦ (More)
• Ars Comb.
• 1997
A d-dimensional Perfect Factor is a collection of periodic arrays in which every k-ary (n1Ã—Â· Â· Â·Ã—nd) matrix appears appears exactly once (periodically). The one dimensional case, with a collection of size one, is known as a De Bruijn cycle. The 1and 2-dimensional versions have proven highly applicable in areas such as coding, communications, and locationâ€¦ (More)