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Infinite Trees with Finite Dimensions
Yusuf Hafidh and Edy Tri Baskoro
Combinatorial Mathematics Research Group
Faculty of Mathematics and Natural Sciences
Institut Teknologi Bandung
Keywords:
Infinite Graph, Tree, Metric Dimension, Partition Dimension, Locating-Chromatic Number.
Abstract:
The properties of graph we consider are metric dimension, partition dimension, and locating-chromatic num-
ber. Infinite graphs can have either infinite or finite dimension. Some necessary conditions for an infinite
graph with finite metric dimension has been studied in 2012. Infinite graphs with finite metric dimension will
also have finite partition dimension and locating-chromatic number. In this paper we find a relation between
the partition dimension (locating chromatic number) of an infinite tree with the metric dimensions of its spe-
cial subtree. We also show that it is possible for an infinite trees with infinite metric dimension to have finite
partition dimension (locating-chromatic number).
1 INTRODUCTION
A graph G is an infinite graph if the number of ver-
tices is infinite. Throughout this paper, we will only
consider connected graphs. The distance from vertex
u to vertex v (d
G
(u, v)) is the number of edges in a
shortest path from u to v. The distance from v ∈ V to
S ⊆ V (d
G
(u, S)) is the minimum distance from u to
all vertices in S. If the context is clear, we simply use
d(u, v) and d(u, S).
Let w be a vertex of graph G, we say that w re-
solves vertex u and vertex v in G if d(w, u) 6= d(w, v).
A set of vertices S is called a resolving set of G if any
two different vertices (u, v) is resolved by some ver-
tices in S. Note that verifying S is a resolving set is
achieved by checking all vertices outside of S. The
metric dimension of G (dim(G)) is the minimum car-
dinality of a resolving set. The coordinate of v (with
respect to S), denoted by r
S
(v), is the vector of dis-
tances from v to vertices in S.
Similarly, a set of vertices S resolves u and v if
d(u, S) 6= d(v,S). Let Π = {π
1
, π
2
, ··· , π
k
} be a par-
tition of V , Π is a resolving partition if and different
vertices u and v is resolved by a partition class in Π.
To verify Π is a resolving partition, we only need to
check vertices in the same partition class. The parti-
tion dimension of G (pd(G)) is the minimum number
of partition classes in a resolving partition. The repre-
sentation of v (r(v|Π)), is the vector of distances from
v to the partition classes in the ordered partition Π.
A map c : V → {1, 2, · · · , k} is a k-coloring if any
two adjacent vertices u and v receive different colors.
A coloring c is a locating k-coloring (or simply locat-
ing coloring) if the partition Π
c
of V induced by c,
is a resolving partition. The locating-chromatic num-
ber of G (χ
L
(G)), is the smallest integer k such that
G has a locating k-coloring. The color code of v is
r
c
(v) = r(v|Π
c
).
Note that any locating coloring will induce a re-
solving partition, therefore we have pd(G) ≤ χ
L
(G).
Throughout this paper, the word dimension corre-
sponds to either metric dimension, partition dimen-
sion, or locating chromatic number.
A well known relation between the metric dimen-
sion of a graph and its partition dimension is given in
the following theorem.
Theorem 1.1. (G. Chartrand and Zhang, 2000) For
any graph G,
pd(G) ≤ dim(G) + 1.
The relation between metric dimension and
locating-chromatic number of a graph is given as fol-
lows.
Theorem 1.2. (G. Chartrand and Zhang, 2002) Let
G be a graph with chromatic number χ(G) (the small-
est positive integer k such that G have a k-coloring).
Then,
χ
L
(G) ≤ dim(G) + χ(G).
Hafidh, Y. and Baskoro, E.
Infinite Trees with Finite Dimensions.
DOI: 10.5220/0009876300002775
In Proceedings of the 1st International MIPAnet Conference on Science and Mathematics (IMC-SciMath 2019), pages 11-13
ISBN: 978-989-758-556-2
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c
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