Local Antimagic Coloring of Graphs - Prof. Slaminmatematika.fmipa.unand.ac.id/images/seminar-workshop/Pewarnaan Dan... · Local Antimagic Coloring of Graphs Prof. Slamin Universitas

Local Antimagic Coloring of Graphs

Prof. Slamin

Universitas Jember

Workshop KK Kombinatorika: Pewarnaan dan Pelabelan pada Graf

Jurusan Matematika Universitas Andalas

Padang, 25 April 2019

Overview

1. Local Antimagic Vertex Coloring

2. Local Antimagic Total Vertex Coloring

3. Super Local Antimagic Total Vertex Coloring

4. Local Antimagic Edge Coloring

5. Conclusion

2

Local Antimagic Vertex Coloring

Introduction

Definition [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]

The local antimagic labeling on a graph G with |V | vertices and |E |edges is defined to be an assignment f : E → {1, 2, · · · , |E |} so that

the weights of any two adjacent vertices u and v are distinct, that is,

w(u) 6= w(v) where w(u) = Σe∈E(u)f (e) and E (u) is the set of edges

incident to u.

• Any local antimagic labeling induces a proper vertex coloring of G

where the vertex u is assigned the color w(u).

• The local antimagic chromatic number, denoted by χla(G ), is the

minimum number of colors taken over all colorings induced by local

antimagic labelings of G .

• For any graph G , χla(G ) ≥ χ(G ) and the difference χla(G )− χ(G )

can be arbitrarily large.

4

Example

Figure 1: The local antimagic vertex coloring of C6 with χla(C6) = 3

5

Cycle

Theorem [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]

For the cycle Cn on n ≥ 3 vertices, χla(Cn) = 3

Proof.

• Label the edges of Cn using the following formula.

f (vivi+1) =

{n − i−1

2 if i is oddi2 if i is even

• The weight of vertices are

w(vi ) =

n if i is odd, i 6= 1

n + 1 if i is even

2n − b n2c if i = 1

• Thus χla(Cn) ≤ 3.

6

Cycle (cont.)

Proof (cont.)

• Suppose n is even and there exists f that induces a 2-coloring of Cn.

• Let x be the color of vi if i is odd and y if i is even.

• Then n2 (x + y) = 2[ n(n+1)

2 ] and hence x + y = 2n + 2.

• If f (vivi+1) = n, then w(vi ) and w(vi+1) are at least n+ 1 and n+ 2.

• Thus x + y ≥ 2n + 3, which is a contradiction.

• So χla(Cn) ≥ 3 if n is even.

• Further, χ(Cn) = 3 if n is odd and hence χla(Cn) ≥ 3.

• Therefore χla(Cn) = 3.

7

Tree


For any tree T with l leaves, χla(T ) ≥ l + 1.

Figure 2: Tree with l leaves

8

Path


For path Pn on n ≥ 3 vertices, χla(Pn) = 3.

Figure 3: The local antimagic vertex coloring of P7 with χla(P7) = 3

9

Friendship graph


For friendship graph Fn with n ≥ 2, χla(Fn) = 3.


For friendship graph Fn with n ≥ 2 by removing an edge e, χla(Fn −{e}) = 3.

Figure 4: The local antimagic vertex coloring of F4 and F4 − {e}10

Complete bipartite graph


For complete bipartite graph Km,n with m, n ≥ 2, χla(Km,n) = 2 if and

only if m ≡ n (mod 2).


For complete bipartite graph K2,n with n ≥ 2, χla(K2,n) = 2 for even

n ≥ 2 and χla(K2,n) = 3 for odd n ≥ 3 or n = 2.

11

Graph Ln


Graph Ln for n ≥ 2 that is obtained by inserting a vertex to each edge

vvi , 1 ≤ i ≤ n − 1, of the star, χla(Ln) = n + 1.

Figure 5: The local antimagic vertex coloring of L7 with χla(L7) = 8

12

Wheel


Wheel Wn of order n + 1 for n ≥ 3, χla(Wn) = 4 if n ≡ 1, 3 (mod 4),

χla(Wn) = 3 if n ≡ 2 (mod 4), and 3 ≤ χla(Wn) ≤ 5 if n ≡ 0 (mod 4).

Figure 6: The local antimagic vertex coloring of W6 with χla(W6) = 3

13

Fan

Corollary

Fan fn of order n + 1 for n ≥ 6 and n ≡ 2 (mod 4), χla(fn) = 3.

Figure 7: The local antimagic vertex coloring of f6 with χla(f6) = 3

14

Graph resulting from operation


For the graph H = G + K2 where G is a graph of order n ≥ 4, then

χla(G ) + 1 ≤ χla(H) ≤

{χla(G ) + 1 for n is even

χla(G ) + 2 otherwise.

15

Kite

Theorem [Nazula, S, Dafik (2018)]

For the kite Ktn,m with n ≥ 3 and m ≥ 1, χla(Ktn,m) = 3.

Proof.

• Label the edges of Ktn,m using the following formula.

f (uiui+1) =

{m+i2 if m + i is even

m+3−i2 + b n2c+ b n−12 c if m + i is odd

f (unu1) =

{b n2c+ m+1

2 if m is odd

b n+12 c+ m

2 if m is even

f (u1v1) =

{m2 if m is evenm+12 + n if m is odd

f (vjvj+1) =

{m−j2 if m + j is even

m+j+12 + n if m + j is odd

16

Kite (cont.)

Proof (cont.)


w(ui ) =

3m2 + 2b n+1

2 c+ b n2c if m is even and i = 1

n + m + 1 if m + i is even

n + m if m + i is odd3m+3

2 + b n2c+ n if m is odd and i = 1

w(vi ) =

{n + m if m + j is even

n + m + 1 if m + j is odd

• Thus χla(Ktn,m) ≤ 3.

• The lower bound uses χla(Cn) due to Arumugam et al.

• Since χla(Cn) = 3 and the kite Ktn,m contains contains a subgraph

that is isomorphic to the Cn, then χla(Ktn,m) ≥ 3.

• Therefore χla(Ktn,m) = 3.

17

Kite (cont.)

Figure 8: The local antimagic vertex coloring of Kt5,6 with χla(Kt5,6) = 3

18

Cycle with two neighbour pendants

Theorem [Nazula, S, Dafik (2018)]

For cycle with two neighbour pendants Cpn with n ≥ 3, χla(Cpn) = 4.

Proof.

• Label the edges of Cpn using the following formula.

f (uiui+1) =

{i+12 if i is odd

n + 1− i2 if i is even

f (unu1) = dn + 1

2e

f (uivi ) = n + i if i = 1, 2


w(ui ) =

b 3n+6

2 c if i = 1

2n + 3 if i = 2

n + 2 if i is odd and i ≥ 3

n + 1 if i is even and i ≥ 4

w(vi ) = n + i if i = 1, 2 19

Cycle with two neighbour pendants (cont.)

Proof (cont.)

• Thus χla(Cpn) ≤ 4.

• To show the lower bound, we suppose that f (u1v1) = m1 and

f (u2v2) = m2.

• Then w(v1) = m1, w(v2) = m2, w(u1) > m1 and w(u2) > m2

• Clearly, w(v1) 6= w(v2).

• As u1 is neighbour of u2, then w(u1) 6= w(u2).

• This implies that χla(Cpn) ≥ 4.

• We conclude that χla(Cpn) = 4.

20

Cycle with two neighbour pendants (cont.)

Figure 9: The local antimagic vertex coloring of Cp6 with χla(Cp6) = 4

21

Corona Product of Graphs

Theorems [Arumugam, Lee, Premalatha, Wang (2018+)]

• χla(Pn � K1) = n + 2 for n ≥ 4.

• χla(Pn � Km) = mn + 2 for m ≥ 2 and n ≥ 2.

• χla(Cn � K1) = n + 2 for n ≥ 4.

• χla(Cn � K2) = 2n + 2 for n ≥ 4.

• χla(C3 � K2) = 3m + 3 for m ≥ 2.

• χla(Cn � Km) = mn + 3 for odd n ≥ 5 except finitely many m?s.

• χla(Kn � K1) = 2n − 1 for n ≥ 3.

• χla(Kn � Km) = mn + n for m ≥ 2 and n ≥ 3.

22

Local Antimagic Total Vertex

Coloring

Local Antimagic Total Vertex Coloring

Definition [Putri, Dafik, Agustin, Alfarisi (2018)]

The local vertex antimagic total labeling on a graph G with |V | vertices

and |E | edges is defined to be an assignment f : V∪E → {1, 2, · · · , |V |+|E |} so that the weights of any two adjacent vertices u and v are distinct,

that is, w(u) 6= w(v) where w(u) = f (u) + Σe∈E(u)f (e) and E (u) is the

set of edges incident to u.

• Any local vertex antimagic total labeling induces a proper vertex

coloring of G where the vertex u is assigned the color w(u).

• The local antimagic total chromatic number, denoted by χlat(G ), is

the minimum number of colors taken over all colorings induced by

local vertex antimagic total labelings of G .

• For any graph G , χlat(G ) ≥ χ(G ).

24

Example

Figure 10: The local antimagic total vertex coloring of C6 with χlat(C6) = 2

25

Particular class of trees

Putri, Dafik, Agustin, Alfarisi (2018)

• For star Sn with n ≥ 2, χlat(Sn) = 2.

• For double star Sn,m with n ≥ 2 and m ≥ 2, χlat(Sn,m) ≤ 3.

• For banana tree Bm,n with n ≥ 3 and m ≥ 3,

χlat(Bm,n) ≤

4, if n odd, m odd

5, if n odd, m even

6, if n even, m even

26

Wheel

Theorem [S, Dafik, Hasan (2018)]

For wheel Wn of order n + 1,

χlat(Wn) =

{3, n ≡ 0 (mod 2)

4, n ≡ 1 (mod 2).

Proof.

• Label all vertices and edges of Wn using the following formula.

f (xixi+1) = i , for 1 ≤ i ≤ n − 1

f (xnx1) = n

f (cxi ) = 2n + 1− i , for 1 ≤ i ≤ n

f (xi ) =

2n + 2, for i = 1

3n, for i = 2, n odd

3n − i + 2− (−1)i+n, untuk 2 ≤ i ≤ n

f (c) = 3n + 1.27

Wheel (cont.)

Proof (cont.).

• The weight of the vertices are

w(xi ) =

5n + 1, for (i + n) even

5n + 2, for i = 2, n odd

5n + 3, for (i + n) odd or i = 1, n odd

w(c) =n(3n + 7)

2+ 1.

• Thus χlat(Wn) ≤ 3 for n even and χlat(Wn) ≤ 4 for n odd.

• Since χlat(Wn) ≥ χ(Wn) = 3 for n even and χlat(Wn) ≥ χ(Wn) = 4

for n odd.

• Therefore, χlat(Wn) = 3 for n even and χlat(Wn) = 4 for n odd.

28

Wheel (cont.)

Figure 11: The local antimagic total vertex coloring of W6 with χlat(W6) = 3

29

Other wheel related graphs

Theorems [S, Dafik, Hasan (2018)]

• For fan Fn of order n ≥ 3, χlat(fn) = 3

• For friendship graph fn of order 2n + 1 with n ≥ 2, χlat(fn) = 3

30

Broom

Theorem [Nikmah, S, Hobri (2018)]

For broom Bn,m with m ≥ 3 and n −m ≥ 2, χla(Bn,m) = n −m + 2

Theorem [Nikmah, S, Hobri (2018)]

For double broom Bn,m with m ≥ 3 and n −m ≥ 2,

1. χlat(Bn,m) ≤ 3, and

2. χlat(Bn,m) = 3, if n >7 + 2m +

√48m − 23

2.

31

Prism and Mobius ladder

Theorem [Hasan, S, Dafik (2018)]

For prism Dn of order 2n with n ≥ 3,

χlat(Dn) =

{2, n ≡ 0(mod 2)

3, n ≡ 1(mod 2).

Theorem [Hasan, S, Dafik (2018)]

For Mobius ladder M2n of order 2n with n ≥ 3,

χlat(M2n) =

{3, n ≡ 0(mod 2)

2, n ≡ 1(mod 2).

32

Super Local Antimagic Total Ver-

tex Coloring

Super Local Antimagic Total Vertex Coloring

Definition

The super local vertex antimagic total labeling on a graph G with |V |vertices and |E | edges is defined as local vertex antimagic total labeling

where the vertices of G receive the smallest labels, that is, 1, 2, ..., |V |.

• Any super local vertex antimagic total labeling induces a proper

vertex coloring of G where the vertex u is assigned the color w(u).

• The super local antimagic total chromatic number, denoted by

χslat(G ), is the minimum number of colors taken over all colorings

induced by super local vertex antimagic total labelings of G .

• For any graph G , χslat(G ) ≥ χlat(G ) ≥ χ(G ).

34

Example

Figure 12: The super local antimagic total vertex coloring of C6 with

χslat(C6) = 2

35

Some results

Theorem [S, Dafik, Hasan (2018+)]

For cycle Cn of order n ≥ 3,

χlas(Cn) =

{3, if n is odd

2, if n is even.


For path Pn of order n ≥ 4, 3 ≤ χslat(Pn) ≤ 4,

36

Gear and Generalised Wheel


For gear Jn,1 of order 2n + 1 and odd n ≥ 5, χslat(Jn,1) = 3.


For generalized wheel W nm of order mn + 1 where m ≥ 1 and n ≥ 3,

χslat(Wnm) =

{3, n even

4, n odd

37

Local Antimagic Edge Coloring

Local Antimagic Edge Coloring

Definition [Agustin, Hasan, Dafik, Alfarisi, Prihandini (2017)]

The local edge antimagic labeling on a graph G with |V | vertices and

|E | edges is defined to be an assignment f : V → {1, 2, · · · , |V |} so

that the weights of any adjacent edges, that is {w(uv) : w(uv) =

f (u) + f (v), uv ∈ E}, are distinct.

• Any local edge antimagic labeling induces a proper edge coloring of

G where the edge uv is assigned the color w(uv).

• The local antimagic edge chromatic number, denoted by χlea(G ), is

the minimum number of colors taken over all colorings induced by

local edge antimagic labelings of G .

• For any graph G , χlae(G ) ≥ χ′(G ).

39

Cycle

Theorem [Agustin, Hasan, Dafik, Alfarisi, Prihandini (2017)]

For the cycle Cn on n ≥ 3 vertices, χlea(Cn) = χla(Cn) = 3

Figure 13: The local antimagic edge coloring of C6 with χlea(C6) = 3

40

Path


For the path Pn on n ≥ 2 vertices, χlea(Pn) = 2.

Figure 14: The local antimagic edge coloring of P7 with χlea(P7) = 2

41

Complete Graph


For the complete graph Kn on n ≥ 3 vertices, χlea(Kn) = 2n − 3

42

Wheel related graphs [Agustin, Hasan, Dafik, Alfarisi, Prihandini (2017)]

• Wheel Wn for n ≥ 3, χlea(fn) = n + 2

• Fan graph fn for n ≥ 3, χlea(fn) = n + 1

• Gear graph Jn,1 for n ≥ 3, χlea(Jn,1) = n + 2

• Helm graph Hn for n ≥ 3, χlea(Hn) = n + 3

• Flower graph Fln for n ≥ 3, χlea(Fln) = 2n + 1

• Sun flower graph SFln for n ≥ 3, χlea(SFln) = 2n − 1

43

Corona Product of Graphs


For corona product of cycle Cn and m copy of K1 where n ≥ 3 and

m ≥ 1, χlea(Cn �mK1) = m + 3,


For corona product of any graph G of order n ≥ 3 and m copy of K1

where m ≥ 1, χlea(G �mK1) = χlea(G ) + m,

44

Local Antimagic Total Edge Coloring

Definition [I.H. Agustin, Dafik, M. Hasan, 2017]

The local edge antimagic total labeling on a graph G with |V | vertices

and |E | edges is defined to be an assignment f : V∪E → {1, 2, · · · , |V |+|E |} so that the weights of any adjacent edges, that is {w(uv) : w(uv) =

f (u) + f (uv) + f (v), uv ∈ E}, are distinct.

• Any local edge antimagic total labeling induces a proper edge

coloring of G where the edge uv is assigned the color w(uv).

• The local antimagic total edge chromatic number, denoted by

χleat(G ), is the minimum number of colors taken over all colorings

induced by local edge antimagic total labelings of G .

• For any graph G , χleat(G ) ≥ χ′(G ).

• If the vertices of G receive the smallest labels, that is, 1, 2, ..., |V |,then the such labeling is called super local edge antimagic total

labeling.

• Any super local edge antimagic total labeling also induces a proper

edge coloring of G . 45

Conclusion

Summary of local antimagic labeling

47

Summary of known results on local antimagic labeling

Other graphs???

48

Open Problems

Open Problem 1

Determine the (super) local antimagic (total) (edge) chromatic number

of other class of graphs G .

Open Problem 2 [Arumugam, Premalatha, Baca, Semanicova-Fenovcikova (2017)]

Characterize the class of graphs G for which χla(G ) = χ(G ).

Open Problem 3

Characterize the class of graphs G for which χlat(G ) = χ(G ), χlea(G ) =

χ′(G ), or χleat(G ) = χ′(G ) and χla(G ) = χlat(G ) = χslat(G ) or

χlea(G ) = χleat(G ) = χsleat(G ).

Open Problem 4

How about local irregular labeling vertex or edge coloring?

49

50

Local Antimagic Coloring of Graphs - Prof. Slaminmatematika.fmipa.unand.ac.id/images/seminar-workshop/Pewarnaan Dan... · Local Antimagic Coloring of Graphs Prof. Slamin Universitas

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