Optimal L(2,1)-Labeling of Triangular Windmill Graphs

Development of an Optimal L(2,1)-Labeling Scheme for
Triangular Windmill Graph
Jazztine Paul Bibit
Maria Christine Kadusale
Marc Christian Satuito
Ma. Jomelyn Ylen
Department of Mathematics
Polytechnic University of the Philippines
March 14, 2017
Jazztine Paul Bibit Maria Christine Kadusale Marc Christian Satuito Ma. Jomelyn Ylen (Department of Mathematics PolytechnicDevelopment of an Optimal L(2,1)-Labeling Scheme for Triangular Windmill GraphMarch 14, 2017 1 / 21

Introduction
L(2,1)-labeling
1 Griggs and Yeh [7] ”Labeling Graphs with a Condition at Distance
Two”
2 Chang and Kuo [5] ”The L(2,1)-labeling Problem on Graphs”
3 Fiala, Klox and Kratochvil [6] ”Fixed Parameter Complexity of
L(2,1)-labelings”
4 Calamoneri and Vocca [4] ”Approximability of the L(h,k)-Labeling
Problem”
5 F. Roberts [10] ”Channel Assignment Problem”

Statement of the Problem
Problem This paper aims to develop an algorithm in labeling all triangular
windmill graph so that it has an optimal L(2, 1)-labeling.

Deﬁnition
Let n ∈ N. The triangular windmill graph W (3, n) is the graph obtained
by taking n copies of K3 with a common vertex such that n ≥ 2.

Deﬁnition
Example
Construct W (3, 3).

Deﬁnition
Example
Construct W (3, 3). This implies n = 3. Thus, we have 3 copies of K3 and
a vertex say j which is adjacent to all vertices in K3.

Deﬁnition
Example
Construct W (3, 3). This implies n = 3. Thus, we have 3 copies of K3 and
a vertex say j which is adjacent to all vertices in K3.
Figure 1. A triangular windmill graph W (3, 3)

L(2,1)-labeling
Deﬁnition
An L(2,1)-labeling of a graph G is a nonnegative integer-valued function
f : V (G) → {0, 1, 2, } such that the following condition holds:

L(2,1)-labeling
Deﬁnition
1 If d(u, v) = 1 then |f (u) − f (v)| ≥ 2, for all u, v ∈ V (G).

L(2,1)-labeling
Deﬁnition
Deﬁnition
A k − L(2, 1)-labeling is an L(2,1)-labeling such that no label is greater
than k.

L(2,1)-labeling
Deﬁnition
An optimal L(2, 1)-labeling of G is a k − L(2, 1)-labeling with minimum k
possible. The maximum label k used by an optimal L(2, 1)-labeling is
called the labeling number of G and denoted by λ(G).

L(2,1)-labeling
Deﬁnition
An optimal L(2, 1)-labeling of G is a k − L(2, 1)-labeling with minimum k
possible. The maximum label k used by an optimal L(2, 1)-labeling is
called the labeling number of G and denoted by λ(G).
Deﬁnition
The L(2, 1)-labeling number λ(G) is the minimum k such that G has an
L(2, 1)-labeling with max{f (v) : v ∈ V (G)} = k.

Example
Consider C3.

Example
Consider C3.
Figure 2. The graph of C3

Example
Consider C3.
Figure 2. The graph of C3
Figure 3. The L(2,1)-labeling of C3

Theorem
Chang-Kuo algorithm is an algorithm for obtaining an L(2,1)-labeling of a
given non-empty graph.

Theorem
Input: A graph G = (V , E)
Output: L(2,1)-labeling of G and k is the maximum label

Theorem
Input: A graph G = (V , E)
Output: L(2,1)-labeling of G and k is the maximum label
Idea: In each step, ﬁnd a maximal 2-stable set from these unlabeled
vertices that are distance at least 2 away from those vertices labeled
in the previous step. Then label all vertices in that 2-stable
set with the index i in current stage. The index i starts from 0 and then
increases by 1 in each step. The maximum label k is the ﬁnal value of i.

Initialization: Set S−1 = ∅; V = V (G); i = 0.

Initialization: Set S−1 = ∅; V = V (G); i = 0. The following steps are as
follows:
S1. Determine Fi and Si where
Fi = {x ∈ V : x is unlabeled and d(x, y) ≥ 2, ∀y ∈ S−1}. Si is a
maximal 2-stable subset of Fi . If Fi = ∅ then set Si = ∅.

follows:
S2. Label these vertices in Si (if there is any) by 1.

follows:
S3. V ← V Si .

follows:
S3. V ← V Si .
S4. V = ∅ , then i ← i + 1; go to Step 1.

follows:
S3. V ← V Si .
S4. V = ∅ , then i ← i + 1; go to Step 1.
S5. Record the current i as k (which is the maximum label).

Algorithm 4.1
Input: The graph of triangular windmill graph W (3, n) where
n ∈ {2, 3, 4, . . .}

Algorithm 4.1
n ∈ {2, 3, 4, . . .}
Output: Optimal L(2, 1)-labeling of W (3, n)
The following steps are as follows:

Algorithm 4.1
n ∈ {2, 3, 4, . . .}
S1. Let W (3, n) be the triangular windmill graph with the vertex set of
V = {v0, v1, . . . , vn, u1, u2, . . . , u2n} and edge set of
E = {v0ui |i = 1, 2, . . . , 2n} ∪ {v0vj |j =
1, 2, . . . , n} ∪ {vku2k−1, vku2k, u2k−1u2k|k = 1, 2, . . . , n}.

Algorithm 4.1
n ∈ {2, 3, 4, . . .}
E = {v0ui |i = 1, 2, . . . , 2n} ∪ {v0vj |j =
S2. Let j ∈{0, 1, 2, . . . , n} and i ∈{1, 2, . . . , 2n}. Also, let
ui , vj ∈ V (W (3, n)). Label the vertices of W(3,n) using f where

Algorithm 4.1
n ∈ {2, 3, 4, . . .}
E = {v0ui |i = 1, 2, . . . , 2n} ∪ {v0vj |j =
S2. Let j ∈{0, 1, 2, . . . , n} and i ∈{1, 2, . . . , 2n}. Also, let
ui , vj ∈ V (W (3, n)). Label the vertices of W(3,n) using f where
f (vj ) =
0, if j = 0,
j + 1, if j = 0.
and f (ui ) =



i + 3
2
+ n, if i is odd,
f (ui−1) + n, if i is even.

Proof
Outline of Proof
We propose the following:
1 All W (3, n) has L(2, 1)-labeling.
2 L(2, 1)-labeling produces distinct labels.
3 Algorithm 4.1 produces optimal L(2, 1)-labeling.

W (3, n) has L(2, 1)-labeling
The proof uses Chang-Kuo Algorithm.
Proposition 4.1
All triangular windmill graph has L(2, 1)-labeling.
Proof.
Since the triangular windmill graph is a non-empty connected graph, then
the algorithm by Chang and Kuo produced L(2, 1)-labeling. Therefore, for
all triangular windmill graph has L(2, 1)-labeling.

Proposition 4.2
L(2, 1)-labeling for all triangular windmill graph W (3, n) will produce
distinct labels.
Proof.
Assume that two vertices, u and v ∈ V , have the same label h. Since the
distance between any two vertices u and v is at most two, then either
d(u, v) = 1 or d(u, v) = 2.

Proposition 4.2
distinct labels.
Proof.
d(u, v) = 1 or d(u, v) = 2. Note that by Deﬁnition of L(2, 1)-labeling.

Proposition 4.2
distinct labels.
Proof.
1 d(u, v) = 1 ⇒|f (u) − f (v)| = |h − h| = 0 < 2.

Proposition 4.2
distinct labels.
Proof.
1 d(u, v) = 1 ⇒|f (u) − f (v)| = |h − h| = 0 < 2.
2 d(u, v) = 2 ⇒|f (u) − f (v)| = |h − h| = 0 < 1.

Proposition 4.2
distinct labels.
Proof.
1 d(u, v) = 1 ⇒|f (u) − f (v)| = |h − h| = 0 < 2.
2 d(u, v) = 2 ⇒|f (u) − f (v)| = |h − h| = 0 < 1. Contradicts the
deﬁnition of L(2,1)-labeling.

Proposition 4.3
Algorithm 4.1 produces optimal L(2,1)-labeling for all triangular windmill
graph.
WNTS the following holds:
1 Algorithm 4.1 produces L(2, 1)-labeling for W (3, n).
a. If d(u, v) = 1, then |f (u) − f (v)| ≥ 2, for any u, v ∈ V .
b. If d(u, v) = 2, then |f (u) − f (v)| ≥ 1, for any u, v ∈ V .
2 The labeling number for W (3, n) is 3n + 1.

Algorithm 4.1 produces L(2,1)-labeling for W(3,n).
Proof for Case a
(a.) If d(u, v) = 1, then |f (u) − f (v)| ≥ 2, for any u, v ∈ V .
Let i ∈ {1, 2, ..., n}, j ∈ {0, 1, 2, ..., n} and k ∈ {1, 2, ..., n}
(i.) v0, ui
(ii.) v0, vj
(iii.) vk, u2k−1
(iv.) vk, u2k
(v.) u2k−1, u2k

Algorithm 4.1 produces L(2,1)-labeling for W(3,n).
Proof for Case b
(b.) If d(u, v) = 2, then |f (u) − f (v)| ≥ 1, for any u, v ∈ V .
(i.) vi , vj where i, j ∈ {1, 2, ..., n} and i = j
(ii.) ui , uj where i = j and i, j ∈ {1, 2, ..., 2n};
if i is even then j = i − 1
if i is odd then j = i + 1
(iii.) vj , ui when i = 2j, i = 2j − 1 and j = {1, 2, ..., n}, i = {1, 2, ..., 2n}

(2.) The labeling number for W (3, n) is 3n + 1.
Proof. Claim that the labeling number of W (3, n) is 3n + 1. Let k be the
maximum label.
1 Assume k < 3n.
2 Assume k = 3n.

(2.) The labeling number for W (3, n) is 3n + 1.
Proof. Claim that the labeling number of W (3, n) is 3n + 1. Let k be the
maximum label.
1 Assume k < 3n.
2 Assume k = 3n.
Contradicts the deﬁnition of L(2,1)-labeling.

Conclusion
The researchers reveal that:
1. All W (3, n) has L(2, 1)-labeling.
2. Algorithm 4.1 produces optimal L(2, 1)-labeling. Moreover, the labeling
number for W (3, n) is 3n + 1.

Recommendation
1 Give a speciﬁc type of graph and give its optimal L(2, 1)-labeling.
2 Let m, n ∈ N where m = n. Observe the L(m, n)-labeling of simple
graphs and develop an algorithm so that simple graphs like path,
cycle and complete graphs has optimal L(m, n)-labeling.

References
1. V. K. Balakrishnan, Schaums Outline of Theory and Problems of
Graph Theory (1997)
2. H. L. Bodlaender, T. Kloks, R. B. Tan and J. Van Leeuwen,
Approximations for Colorings of Graphs. ”The Computer Journal,47”
(2004), 193-204.
3. T. Calamoneri, The L(h, k)-Labelling Problem: A Survey and
Annotated Bibliography, ”The Computer Journal, 49”, (2006)
585-608.
4. T. Calamoneri and P. Vocca,Approximability of the L(h, k)-Labelling
Problem. ”Proceedings of 12th Colloquium on Structural Information
and Communication Complexity,Le Mont Saint-Michel, France, 24-26
May, 65-77, Lecture Notes in Computer Science 3499, Springer
Verlag, Berlin.”
5. Gerard J. Chang and David Kuo, The L(2,1)-Labeling Problem on
Graphs, (1994)
6. Fiala J, Kloks T, Kratochvl J. Fixed parameter complexity of
-labelings. ”Discrete Applied Mathematics,” (2001) 113, 5972.Jazztine Paul Bibit Maria Christine Kadusale Marc Christian Satuito Ma. Jomelyn Ylen (Department of Mathematics PolytechnicDevelopment of an Optimal L(2,1)-Labeling Scheme for Triangular Windmill GraphMarch 14, 2017 20 / 21

References
7. J. R. Griggs and R. K. Yeh, Labeling graphs with a condition at
distance two, ”SIAM J. Discrete Math., 5” (1992), 586-595.
8. M. M. Halldorsson, Approximating the L(h, k)-labeling problem.
”International Journal of Mobile Network Design and Innovation,
1(2)” (2006), 113-117.
9. Andrew Lum, Upper Bounds on the L(2,1)-Labeling Number Of
Graphs With Maximum Degree, (2007)
10. F. Roberts, Private Communication of Roberts to Griggs Yeh
11. Zhendong Shao, The Research on the L(2,1)-labeling problem from
Graph theoretic and Graph Algorithmic Approaches, (2012)

Optimal L(2,1)-Labeling of Triangular Windmill Graphs

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