(2,1)-TOTAL NUMBER OF JOINS OF PATHS AND CYCLES
Weifan Wang, Jing Huang, Danjun Huang, Haina Sun · Taiwanese Journal of Mathematics · 2012
The $(2,1)$-total number $\lambda_2^t(G)$ 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.