In his 1992 Ph.D. thesis Chang identified an efficient way to dominate m-by-n grid graphs and conjectured that his construction gives the most efficient dominating sets for relatively large grids. In 2011 Goncalves, Pinlou, Rao, and Thomasse proved Chang's conjecture, establishing a closed formula for the domination number of a grid. In March 2013, Fata, Smith and Sundaram established upper bounds for the k-distance domination numbers of grid graphs by generalizing Chang's construction of dominating sets to k-distance dominating sets. In this paper we use algebraic and geometric arguments to improve the upper bounds established by Fata, Smith, and Sundaram for the k-distance domination numbers of grids.

Author Bio

Michael Farina graduated cum laude from Florida Gulf Coast University in 2015, where he majored in mathematics. During his time at FGCU he participated in this paper, mostly over the summers. He is currently working as a mathematical statistician and plans on continuing his graduate studies in statistics in the future.

Armando Grez graduated from Florida Gulf Coast University with a B.S. in Mathematics and a Minor in Statistics. This research was conducted during his last two years under the guidance of Dr. Erik Insko. He has presented this work at the 2014 MAA Mathfest in Portland, Underrepresented Students in Topology and Algebra Research Symposium at UC-Berkeley, and Embry-Riddle Undergraduate Mathematics Conference in Daytona. He is currently pursuing a Ph.D. in Mathematics at Iowa State University.