PC-Trees and Planar Graphs

44
Distance labeling with non-monotonic constraint
AUTHOR: Chia-Hung Hsia, Roger K. Yeh 葉光清
Department of Applied Mathematics, Feng Chia University, Taichung 407, Taiwan
rkyeh@math.fcu.edu.tw
ABSTRACT
A distance labeling of a graph is a mapping from the vertex set of that graph to the
real line such that the (Euclidean) distance between two points depends on the distance
between their pre-images in the graph with given conditions. In particular, we define
the following labeling. Given nonnegative reals δ1, δ2, δ3, an L(δ1, δ2, δ3)-labeling ( or a
labeling with constraint (δ1, δ2, δ3) ) of a graph G is a function from V (G) to the real line
R such that |f(u), f(v)| ≥ δi whenever dG(u, v) = i for 1 ≤ i ≤ 3, where dG(u, v) is
the distance between u and v in G. Notice that |f(u) − f(v)| is the Euclidean distance
between f(u) and f(v) in R.
The span of a labeling f is the difference of the maximum value and the minimum value
of f on V (G). The L(δ1, δ2, δ3) number ( or L(δ1, δ2, δ3)-span ) of G is the infimum span
over all L(δ1, δ2, δ3)-labelings. It can be shown that the infimum span is attainable, that
is, a minimum.
We say the L(δ1, δ2, δ3)-span is monotone if λ(G; δ1, δ2, δ3) ≥ λ(H; δ1, δ2, δ3) for all subgraphs
of a graph G; and is hereditary if the inequality is true for every induced subgraphs
H of G. We show that if δ2 ≥ δ3 then the L(δ1, δ2, δ3)-span is hereditary. It is also monotone
provided that δ1 ≥ δ2. However, some examples tell us that if δ1 < δ2 then it is
not monotone. That is the sequence < δ1, δ2, δ3 > is not monotonic. In the ordinary
graph coloring, this kind of property of monotone seems obviously true, for example
the chromatic number or the chromatic index but not in the distance labeling problem.
Thus we are interested in this type of distance labelings. This article will consider the
L(1, d, 1)-labeling for real d ≥ 1.
KEYWORDS: Graph coloring, distance.