(2,1)-Total labelling of trees with sparse vertices of maximum degree
The (2, 1)-total labelling number lambda(t)(2) (G) of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G Such that no two adjacent vertices, or two adjacent edges, have the same label and the difference between the labels of a vertex and its incident edges is at least 2. Let T be a tree with maximum degree Delta >= 4. Let D(Delta)(T) denote the set of integers k for which there exist two distinct vertices of maximum degree of distance at k in T. It was known that Delta + 1  = 3, there exist infinitely many trees T with Delta >= 4 and k is not an element of D(Delta)(T) such that lambda(t)(2)(T) = Delta + 2. (C) 2008 Elsevier B.V. All rights reserved.
(2,1)-total labelling tree maximum degree combinatorial problems