Radio Number for 5th Power of Paths
Authors:
Alberto Acevedo, Blair Blokzyl, Krista Leal, Joana Luna, Samuel MarrujoMentor:
Min-Lin Lo, Associate Professor of Mathematics, California State University San BernardinoLet G be a connected graph. For any two vertices u and v, let d(u, v) denote the distances between u and v in G. The maximum distance between any pair of vertices is called the diameter of G and denoted by diam(G). A radio labeling (or multi-level distance labeling) of a connected graph G is a function f : V (G) -> {0,1,2,3... } with the property that jf(u) f(v)j diam(G) d(u, v)+1 for every two distinct vertices u and v of G. The span of f is defined as maxu;veV (G) fjf(u) f(v)jg. The radio number of G is the minimum span over all radio-labelings of G. The 5th power of G is a graph constructed from G by adding edges between vertices of distance five or less apart in G. The radio number for paths, square paths, and cube paths were solved. In this presentation we will discuss the progress we made towards finding the radio number for 5th power of paths during a 2012 MAA summer research program funded by NSA (grant H98230-11-1-0215) and NSF (grants DMS-0845277 and DMS-1156582).