The L(h, 1, 1)-labelling problem for trees

Let h>=1 be an integer. An L(h,1,1)-labelling of a (finite or infinite) graph is an assignment of nonnegative integers (labels) to its vertices such that adjacent vertices receive labels with difference at least h, and vertices distance 2 or 3 apart receive distinct labels. The span of such a labelling is the difference between the maximum and minimum labels used, and the minimum span over all L(h,1,1)-labellings is called the @l"h","1","1-number of the graph. We prove that, for any integer h>=1 and any finite tree T of diameter at least 3 or infinite tree T of finite maximum degree, max{max"u"v"@?"E"("T")min{d(u),d(v)}+h-1,@D"2(T)-1}@[email protected]"h","1","1(T)@[email protected]"2(T)+h-1, and both lower and upper bounds are attainable, where @D"2(T) is the maximum total degree of two adjacent vertices. Moreover, if h is less than or equal to the minimum degree of a non-pendant vertex of T, then @l"h","1","1(T)@[email protected]"2(T)+h-2. In particular, @D"2(T)[email protected][email protected]"2","1","1(T)@[email protected]"2(T). Furthermore, if T is a caterpillar and h>=2, then max{max"u"v"@?"E"("T")min{d(u),d(v)}+h-1,@D"2(T)-1}@[email protected]"h","1","1(T)@[email protected]"2(T)+h-2 with both lower and upper bounds achievable.