Two papers co-authored by Assistant Professor of Mathematics Courtney Gibbons were recently published.
“Critical Pebbling Numbers of Graphs” appeared in the November issue of the Journal of Combinatorial Mathematics and Combinatorial Computing.
Co-authored with Josh Laison of Willamette University and Erick Paul of the University of Illinois at Urbana-Champaign, the paper defines three new pebbling parameters of a connected graph G, the r-, g-, and u-critical pebbling numbers.
Together with the pebbling number, the optimal pebbling number, the number of vertices n and the diameter d of the graph, this yields 7 graph parameters. The authors determined the relationships between these parameters. They also investigated properties of the r-critical pebbling number and distinguished between greedy graphs, thrifty graphs and graphs for which the r-critical pebbling number is 2^d.
Another paper, “Rational Combinations of Betti Diagrams of Complete Intersections,” appears online in the Journal of Algebra and Its Applications. Gibbons wrote the paper with three former students from the Willamette University Mathematics Consortium research experience for undergraduates (REU). The students are all now in mathematics Ph.D. programs.
The group investigated decompositions of Betti diagrams over a polynomial ring within the framework of Boij-Soederberg theory. To determine if it is possible to decompose a given Betti diagram into the Betti diagrams of complete intersections, they determined the extremal rays of the cone generated by the diagrams of complete intersections and provided a rudimentary algorithm for decomposition.
This work adds to the relatively young field of Boij-Soederberg theory, a subfield of commutative algebra.