(2,1)-Total number of joins of paths and cycles

摘要

The (2, 1)-total number λ~t_2(G) of a graph G is the width of the smallest range of integers that suffices to label the vertices and edges of G such that no two adjacent vertices or two adjacent edges have the same label and the difference between the label of a vertex and its incident edges is at least 2. In this paper, we characterize completely the (2, 1)-total number of the join of two paths and the join of two cycles.