Key Generation in Cryptography Using Radio Path Coloring

An L(p_{1},~p_{2},~p_{3},~{\dots },~p_{m}) - labeling of a graph G is an assignment of positive integers to the vertices of G such that the difference in the labels assigned to the vertices at distance i should be at least p_{i} . The particular case of p_{1} = d, \,\,p_{2} = d-1, \,\,p_{3} = d-2, {\dots }, p_{d} = 1 where d is the diameter of the graph, was known as the radio labeling of G . The minimum value of the maximum integer used in any feasible radio labeling of G was called as radio number of G denoted by rn(G) . The idea of radio path coloring was conceived to ensure secure communication in networks. If there exists a path between each pair of vertices, such that, the labeling in that path is an L(p_{1},\,\,p_{1}-1,\,\,p_{1}-2,\,\,{\dots },\,\,1)- labeling, then such a labeling was called an L(p_{1},\,\,p_{1}-1,\,\,p_{1}-2,\,\,{\dots },\,\,1)- path coloring or a radio path coloring of G . The minimum value of the largest label used in such a coloring was called as the radio path connection number. Earlier researchers have studied the case of p_{1}=2 for different classes of graphs. We focus on the more general case of p_{1} \geq 3 and obtain an upper bound on the radio connection number k_{p_{1}c}(G) , of any semi-Hamiltonian graph