Chung defined a pebbling move on a graph G to be the removal of two pebbles from one vertex and the addition of one pebble to an adjacent vertex. The pebbling number of a connected graph is the smallest number f (G) such that any distribution of f (G) pebbles on G allows one pebble to be moved to any specified, but arbitrary vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphs G
Graph pebbling is the study of moving discrete pebbles from certain initial distributions on the vertices of a graph to various target distributions via pebbling moves. A pebbling move removes two pebbles from a vertex and places one pebble on one of its neighbors (losing the other as a toll).For t ≥ 1 the t-pebbling number of a graph is the minimum number of pebbles necessary so that from any initial distribution of them it is possible to move t pebbles to any vertex.We provide the best possible upper bound on the t-pebbling number of a diameter two graph, proving a conjecture of Curtis, et al., in the process. We also give a linear time (in the number of edges) algorithm to t-pebble such graphs, as well as a quartic time (in the number of vertices) algorithm to compute the pebbling number of such graphs, improving the best known result of Bekmetjev and Cusack.Furthermore, we show that, for complete graphs, cycles, trees, and cubes, we can allow the target to be any distribution of t pebbles without increasing the corresponding t-pebbling numbers; we conjecture that this behavior holds for all graphs.Finally, we explore fractional and optimal fractional versions of pebbling, proving the fractional pebbling number conjecture of Hurlbert and using linear optimization to reveal results on the optimal fractional pebbling number of vertex-transitive graphs.
Writing in Computer Science education is typically writing to communicate to a professional audience-also known as "writing in the discipline." (WID) A few Computer Science educators have promoted "writing to learn" (WTL) for active learning. A gap exists between these two forms of writing that inhibits the general adoption of writing in Computer Science. We propose that "bridging" informal WTL assignments to formal disciplinary writing as a way of promoting general adoption of writing across all courses, thus improving thinking and writing skills for all Computer Science students. We include examples of assignments that bridge writing to learn and writing in the discipline.
A pebbling move on a connected graph G consists of removing two pebbles from some vertex and adding one pebble to an adjacent vertex. We define f t (G) as the smallest number such that whenever f t (G) pebbles are on G, we can move t pebbles to any specified, but arbitrary vertex. Graham conjectured that f 1
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.