Optimal L(3,2,1)-labeling of trees

Abstract

Given a graph G, an L ( 3 , 2 , 1 ) -labeling of G is an assignment f of non-negative integers (labels) to the vertices of G such that | f ( u ) − f ( v ) | ≥ 4 − i if dist ( u , v ) = i (i = 1, 2, 3). For a non-negative integer k, a k- L ( 3 , 2 , 1 ) -labeling is an L ( 3 , 2 , 1 ) -labeling such that no label is greater than k. The L ( 3 , 2 , 1 ) -labeling number of G, denoted by λ 3 , 2 , 1 ( G ) , is the smallest number k such that G has a k- L ( 3 , 2 , 1 ) -labeling. Chia proved that the L ( 3 , 2 , 1 ) -labeling number of a tree T with maximum degree Δ can have one of three values: 2 Δ + 1 , 2 Δ + 2 and 2 Δ + 3 . This paper gives some sufficient conditions for λ 3 , 2 , 1 ( T ) ≥ 2 Δ + 2 and λ 3 , 2 , 1 ( T ) = 2 Δ + 3 , respectively. As a result, the L ( 3 , 2 , 1 ) -labeling numbers of complete m-ary trees, spiders and banana trees are completely determined.