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

Abstract

Graph labeling problemput nonnegative integers to the vertex with some restrictions. L(h, k) labeling is one kind of graph labeling where adjacent nodes get the value difference 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 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 1988 gives the idea of frequency assignment problem with the restriction “close” and “very close”, where “close” node received frequency that is different and “very close” node received frequency is two or more apart, which 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, the 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 $\lambda _{2,1}(G)$. Here, we propose a polynomial time algorithm to label the graph obtained by the Cartesian product between complete bipartite graph and path. We design the algorithm in such a way that gives exact L(2, 1) labeling of the graph $G=(K_{m,n}\times P_r)$ for the bound of $m,n>5$ and which is $\lambda _{2,1}(G)= m+n$. Our proposed algorithm successfully follow the conjecture of Griggs and Yeh. Finally, we have shown that L(2, 1) labeling of the above graph can be solved in polynomial time for some bound.