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
Copyright
c
 2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
11

2 INFINITE GRAPH WITH
FINITE DIMENSIONS
2.1 Metric dimension
C
´
acares and Puertas (2012) showed that any infinite
graph with bounded degree always has finite metric
dimension.
Theorem 2.1. (J. C
´
aceres and Puertas, 2012) If G is
an infinite graph with maximum degree ∆ and m(≥
1) vertices of degree at least 3, then dim(G) is finite.
Moreover dim(G) ≤ m∆.
The converse of this theorem is not true in general
but it is true for any tree.
Theorem 2.2. (J. C
´
aceres and Puertas, 2012) If an
infinite tree has finite metric dimension, then the set of
vertices of degree at least 3 is finite and has bounded
degree.
These two theorems give a characterization for in-
finite trees to have finite metric dimension.
Corollary 2.1. Let T be an infinite tree, then dim(T )
is finite if and only if T has bounded degree and the
number of vertices of degree at least 3 is finite.
Let T be a tree other than a path, a branch is a
vertex with degree more than two. An end path is
a path connecting a leaf (vertex of degree one) to its
nearest branch. A major branch is a branch with at
least one end path. The metric dimension of a tree is
given in the following theorem.
Theorem 2.3. (Slater, 1975) Let T be a finite tree with
b ≥ 1 major branches v
1
, ··· , v
b
and l leaves. If k
i
is the number of end paths from v
i
, then dim(T ) =
l − b =
∑
b
i=1
(k
i
− 1).
The previous theorem

dim(T ) =
∑
b
i=1
(k
i
− 1)

is
also true for any infinite graph, but we need to gener-
alize the definition of an end path to also be an infinite
ray from a branch, (J. C
´
aceres and Puertas, 2012).
2.2 Partition dimension and
locating-chromatic number
In this section we find a relation between the parti-
tion dimension and locating-chromatic number of an
infinite tree with the metric dimensions of its special
subtree. This show that it is possible for an infinite
trees with infinite metric dimension to have finite par-
tition dimension (locating-chromatic number).
Let T be any infinite tree, define [T ] as the tree
obtained by the following way : for every major
branch v, every end path starting from v is contracted
into one vertex, see figure 1. Note that [T ] is a subtree
Graph T
Graph [T]
Figure 1: Graph T and [T ]
of T .
Theorem 2.4. If T be an infinite tree with bounded
degree. If the maximum number of end path from a
branch is κ, then
pd(T ) ≤ dim([T ])+ κ + 1.
Proof. We will construct a resolving partition for T
with cardinality dim([T ]) + κ + 1. Consider the fol-
lowing algorithm:
1. Let W = {w
1
, w
2
, ··· , w
dim([T ])
} be a resolving set
for subtree [T ].
2. For every major vertex v
i
of T (i = 1, 2, · · · ), let
P
i1
, P
i2
, ··· be the end paths from v
i
.
3. Define Π = {π
1
, π
2
, ··· , π
dim([T ])+κ+1
} with:
• π
i
= {w
i
} for i = 1, 2, ··· , dim([T ]);
• π
i+dim([T ])
= ∪
j
(V (P
ji
) −{v
j
}) for
i = 1, 2, ··· , κ; and
• π
dim([T ])+κ+1
= V (T )−

π
1
∪ ···∪ π
dim([T ])+κ

.
Let u and v be any two different vertices in the
same partition. If u, v ∈ π
dim([T ])+κ+1
, then u, v ∈
V ([T ]), which means that w
i
will resolve u and v for
some i, and so π
i
= {w
i
} also resolve u and v.
If u and v in π
i
with dim([T ])+1 ≤ i ≤ dim([T ])+
κ, let u
0
and v
0
be the nearest branch form u and v,
respectively. If d(u, u
0
) = d(v, v
0
), then the vertex w
i
in W that distinguishes u
0
and v
0
will also distinguish
u and v, this implies that π
i
will distinguishes u and
v. If d(u, u
0
) < d(v, v
0
) then the partition containing
u
0
will distinguish u and v, a similar argument can be
applied if d(u, u
0
) > d(v, v
0
).
So every pair of vertices is resolved by some par-
tition class, therefore Π is a resolving partition, and
pd(T ) ≤ dim([T ])+ κ + 1.
IMC-SciMath 2019 - The International MIPAnet Conference on Science and Mathematics (IMC-SciMath)
12

Theorem 2.5. Let T be an infinite tree with bounded
degree. If the maximum number of end path from a
branch is κ, then
χ
L
(T ) ≤ dim([T ])+ κ + 2.
Proof. First, we will construct a resolving coloring
for T with cardinality dim([T ]) + κ+ 2. Consider the
following algorithm:
1. Let W = {w
1
, w
2
, ··· , w
dim([T ])
} be a resolving set
for subtree [T ].
2. For every major vertex v
i
of T (i = 1, 2, · · · ), let
P
i1
, P
i2
, ··· be the end paths from v
i
.
3. Define the following coloring on V (T):
• Fix a vertex v in T , for every vertex in V (T ),
if the vertex has odd distance to v, color that
vertex with dim([T ]) + κ + 1, otherwise color
the vertex with dim([T ]) + κ +2;
• Recolor w
i
with i for i = 1, 2, · · · , dim([T ]);
• For every vertex in P
i j
, if the distance to its
nearest branch is odd, recolor the vertex with
dim([T ]) + j.
Now we prove that c is a resolving coloring. Let
u and v be two different vertices and assume that
r
c
(u) = r
c
(v). Since W is a resolving set for [T ], if
r
W
(u) = r
W
(v), then u and v must be in two different
end paths from the same major branch, lets say u in
P
i j
and v in P
ik
. But that means the color j and k will
distinguish u and v.
Therefore c is a resolving coloring, and χ
L
(T ) ≤
dim([T ]) + κ + 2.
The previous two theorems, together with corol-
lary 2.1 give a sufficient condition for infinite trees to
have finite partition dimension and locating chromatic
number.
Corollary 2.2. Let T be a tree with bounded degree
and [T ] only have a finite number of branches, then
pd(T ) and χ
L
(T ) are finite.
Note that Theorem 2.4 and Theorem 2.5 also work
for finite graphs.
Corollary 2.3. Let T be a tree (finite or infinite). If
the maximum number of end path from a branch is γ,
then
pd(T ) ≤ dim([T ])+ γ + 1
and
χ
L
(T ) ≤ dim([T ])+ γ + 2.
Remark 2.1. For an infinite tree T with pd(T ) or
χ
L
(T ) finite, it is not necessary that dim(T ) is finite.
For example, if T is an infinite comb (a tree obtained
by attaching one pendant to each vertex in an infinite
path), then dim(T ) is infinite (J. C
´
aceres and Puer-
tas, 2012). Since [T ] is an infinite path with metric
dimension two, we have pd(T ) ≤ 4 and χ
L
(T ) ≤ 5 by
Theorem 2.4 and Theorem 2.5.
ACKNOWLEDGMENTS
This research has been supported by World Class Re-
search Grant, Ministry of Research, Technology and
Higher Education, Indonesia.
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Slater, P. J. (1975). Leaves of trees. Proc. 6th Southeastern
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Infinite Trees with Finite Dimensions
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