On (s,t)-relaxed L(1,1)-labelling of trees

Abstract

ABSTRACTSuppose G is a graph. Let u be a vertex of G. A vertex v is called an i-neighbour of u if. A 1-neighbour of u is simply called a neighbour of u. Let s and t be two non-negative integers. Suppose f is an assignment of non-negative integers to the vertices of G. If, for any vertex u of G, the number of neighbours v (resp. 2-neighbours w) of u with (resp. ) is at most s (resp. t), then f is called an-relaxed -labelling of G. The span of f, denoted by , is the difference between the maximum and minimum integers used by f. The minimum span of -relaxed-labellings of G is called the -relaxed -labelling number of G, denoted by . It is clear that is the so called -labelling number of G. This paper studies the-relaxed -labellings of trees. Let T be a tree with maximum degree Δ. It is proved that and both lower and upper bounds are attainable. For s=0 and fixed, a polynomial-time algorithm is designed to determine if equals the lower bound.