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

Abstract

Graph labeling problem put 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 λ 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 × P r ) for the bound of m, n > 5 and which is λ 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.

Certain Properties of Vague Graphs with a Novel Application

“…Sunitha et al [33][34][35] presented new concepts for fuzzy graphs. Ghosh et al [19,36] investigated new concepts of the signed product and the total signed product on complex graphs. In 2008, Gani and Latha [37] investigated the properties of NIFGs and HIFGs.…”

Efficient Algorithm For L(3,2,1)-Labeling of Cartesian Product Between Some Graphs

“…The above conjecture of Griggs and Yeh [2] worked for the set of graphs like path [2], wheel [2], cycle [2], trees [2,5], co-graphs [5], interval graphs [5], chordal graphs, permutation graph [6], circular arc graph, Cartesian product of complete bipartite graph, path and cycle [14,15,16] etc. The bound λ 2,1 (G) can be computed systematically for some graphs like cycle, path, tree [2,5].…”