L(2,1)-labeling of some zero-divisor graphs associated with commutative rings

Let g = (V, 6) be a simple graph, an L(2,1)-labeling of g is an assignment of labels from non-negative integers to vertices of g such that adjacent vertices get labels which differ by at least by two, and vertices which are at distance two from each other get different labels. The lambda-number of g, denoted by lambda(g), is the smallest positive integer P such that g has an L(2,1)-labeling with all labels as members of the set {0,1, ... ,P}. The zero-divisor graph of a finite commutative ring R with unity, denoted by Gamma(R), is the simple graph whose vertices are all zero divisors of R in which two vertices u and v are adjacent if and only if uv = 0 in R. In this paper, we investigate L(2,1)-labeling of some zero-divisor graphs. We study the partite truncation, a graph operation that allows us to obtain a reduced graph of relatively small order from a graph of significantly larger order. We establish the relation between lambda-numbers of the graph and its partite truncated one. We make use of the operation partite truncation to contract the zero-divisor graph of a reduced ring to the zero-divisor graph of a Boolean ring.