Exact Algorithm for L(2, 1) Labeling of Cartesian Product Between Complete Bipartite Graph and Cycle

Abstract

L(h, k) labeling is one kind of graph labeling where adjacent nodes get the value differ by at least h and the nodes which are at 2 distance apart get value differ by at least k, which has major application in radio frequency assignment, where the assignment of frequency to each node of radio station in such a way that adjacent station get frequency which does not create any interference. Robert in 1998 gives the direction to introduce L(2, 1) labeling. L(2, 1) labeling is a special case of L(h, k) labeling, where the value of h is 2 and value of k is 1. In L(2, 1), labeling difference of label is at least 2 for the vertices which are at distance one apart and label difference is at least 1 for the vertices which are at distance two apart. The difference between minimum and maximum label of L(2, 1) labeling of the graph $G=(V,E)$ is denoted by $\lambda _{2,1}(G)$. In this paper, we propose a super-linear time algorithm to label the graph obtained by the Cartesian product between complete bipartite graph and cycle. We design the algorithm in such a way that gives exact labeling of the graph $G=(K_{m,n}\times C_r)$ for the bound of $m,n>5$ and which is $\lambda _{2,1}(G)= m+n$. Finally, we have shown that L(2, 1) labeling of the above graph can be solved in polynomial time for some bound.