Facial [formula omitted]-edge-labelings of trees

Abstract

Let G be a plane graph. A facial path of G is any path which is a consecutive part of the boundary walk of a face of G. Two edges e1 and e2 of G are facially adjacent if they are consecutive on a facial path of G. Two edges e1 and e3 are facially semi-adjacent if they are not facially adjacent and there is a third edge e2 which is facially adjacent with both e1 and e3, and the edges e1,e2,e3 are consecutive (in this order) on a facial path. An edge-labeling of G with labels 1,2,…,k is a facial L(2,1)-edge-labeling if facially adjacent edges have labels which differ by at least 2 and facially semi-adjacent edges have labels which differ by at least 1. The minimum k for which a plane graph admits a facial L(2,1)-edge-labeling is called the facial L(2,1)-edge-labeling index.In this paper, we prove that the facial L(2,1)-edge-labeling index of any tree T is at most 7; moreover, this bound is tight. In the case when T has no vertex of degree 3 the upper bound for this parameter is 6, which is also tight. If T is without vertices of degree 2 and 3, then its facial L(2,1)-edge-labeling index is at most 5; moreover, this bound is also tight. Finally, we characterize all trees having facial L(2,1)-edge-labeling index exactly 4.