Journal of Information and Computational Science ISSN: 1548-7741
On harmonious coloring and b-coloring of some special graphs of Dutch windmill graph and sunlet graph
S. Blessy1, Dr.A. Arokia Lancy2 1PG Scholar 2Assistant Professor Department of Mathematics, Nirmala College for Women, Coimbatore, TamilNadu, India. [email protected]
Abstract
A proper coloring of a graph is assigning colors to vertices and edges with distinct colors. The harmonious coloring and b-coloring are both proper vertex coloring. In this paper we find the harmonious chromatic number of some special graphs of Dutch windmill graph and b-chromatic number of some special graphs of sunlet graph.
Keywords: Proper coloring, harmonious coloring, harmonious chromatic number , b- coloring, b-chromatic number , Dutch windmill graph, sunlet graph, line graph, middle graph, subdivision graph.
1. INTRODUCTION
All graphs considered in this paper are non-trivial, simple, connected and undirected. Let G be a graph with vertex set V and edge set E. A k-coloring of a
graph G is a partition P={v1, v2, …. vk} of V into independent sets of G. The minimum cardinality k for which G has a k-colouring is the chromatic number 휒(퐺) of graph G.
Harmonious coloring was first proposed by Frank Harary and M.J. Plantholt [4] in 1982. However, this was originally introduced by Hopcraft and Krishnamoorthy [7] in 1983.
The b-chromatic number 휑(퐺) of a graph G is the largest positive integer k such that G admits a proper k-coloring in which every color has a representative adjacent to atleast one vertex in each of the other color classes. Such coloring is called a b- coloring. The b-chromatic number was first introduced by Irving and Manlove in 1999.
The b-chromatic number of corona graphs was proposed by Vernold. V.J., Venkatachalam. M in 2007 [16]. Irving and Manlove [8] have also proved the upper bound of b-chromatic number of a graph G as
φ(G) ≤ ∆(G)+1 ______(1.1)
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The lowerbounds and upperbounds for b-chromatic number of the cartesian product of two graphs were investigated in the work of Kouider M and Maheo [10].
The b-chromatic number for Cartesian product of some families of graph were found by Balakrishnan .R., Francis Raj and Kavaskar. T[1].
Then Vaidya. S.K and Shukla. M [15] found the b-chromatic number of some cycle related graphs.
In this paper, we examine the b-chromatic number of some special graphs of sunlet graph and the harmonious chromatic number of some special graphs of Dutch windmill graph.
2. PRELIMINARIES
Definition 2.1 [3]
A graph G(V,E) is a set of vertices V={푣1, 푣2, … 푣푛} and a set of edges E={푒1, 푒2, … 푒푛}
Definition 2.2 [3]
A vertex is a point where multiple lines meet. It is also called a node. A vertex is denoted by an alphabet. The degree of a vertex is the number of edges which touches the vertex. The minimum vertex degree in a graph G is denoted by 훿(퐺) and the maximum vertex degree is denoted by ∆(퐺)
Definition 2.3 [3]
An edge is a line that connects two vertices. Many edges can be formed from a single vertex.
Definition 2.4 [3]
A vertex with degree one is called pendant vertex. The edge of a graph is said to be pendant if one of its vertices has degree one.
Definition 2.5 [3]
In a graph, two vertices are said to be adjacent, if there is an edge between the two vertices . In a graph, two edges are said to be adjacent, if there is a common vertex between the two edges.
Definition 2.6 [3]
A graph is said to be connected, if for every u,v in G there exists a uv-path in G
A graph is said to be disconnected, if it does not contain atleast two connected vertices.
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Definition 2.7 [3]
A cycle graph or a circular graph is a graph that consists of a single cycle, or some number
of vertices connected in a closed chain. The cycle graph with n vertices is denoted as 퐶푛
Definition 2.8 [16]
The n-sunlet graph is the graph on 2n vertices obtained by attaching n pendant edges to a
cycle graph 퐶푛. These graphs are also known as crown graphs. It is denoted by Sn
Definition 2.9 [2]
The Dutch-Windmill graph is the graph obtained by taking n-copies of the cycle 퐶3 with a 푛 vertex in common. This graph is also known as friendship graph. It is denoted by 퐷3
Definition 2.10 [6]
The line graph of a graph G denoted by L(G) is the graph such that
(i) V[L(G)]=E(G)
(ii) x~y in L(G) if x~y in G for x and y are in E(G). It is denoted by L(G)
Definition 2.11 [12]
The middle graph of a graph G is a graph whose vertex set is V(G)∪E(G) and two vertices x,y in the vertex set of middle graph are adjacent if one of the following case holds:
(i) x,y are in E(G) and x,y are adjacent in G (ii) x is in V(G), y is in E(G) and x,y are incident in G It is denoted by M(G)
Definition 2.12 [9]
The subdivision graph of a graph G is the graph obtained from G by replacing each of its edge by a path of length two, or, equivalently, by inserting an additional vertex into each edge of G. It is denoted by S(G)
Definition 2.13 [5]
A proper vertex coloring of G is the assignment of colors to the vertices of G such that no two adjacent vertices have the same color.
The chromatic number of G is the minimum integer 휒(퐺) such that G has a proper coloring with 휒(퐺) colors.
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Definition 2.14 [4]
A harmonious coloring is a proper vertex coloring such that each pair of colors appears atmost in one edge or in one pair of adjacent vertices.
The harmonious chromatic number of G is the least number of colors in a harmonious
coloring of G. It is denoted by 휒퐻(퐺)
Definition 2.15 [8]
A b-coloring is the proper vertex coloring in which every color class contain a vertex adjacent to every other color class.
The b-chromatic number of a graph G is the maximum number of colors for which G has a proper coloring such that every color class contain a vertex adjacent to every other color class. It is denoted by φ(G)
3. Harmonious coloring
Theorem: 3.1
푛 For every Dutch windmill graph 퐷3 , the harmonious chromatic number is 풏 흌푯(푫ퟑ )=2n+1 where n is the number of copies of 퐶3
Proof:
푛 Let 퐷3 be the Dutch windmill graph.
1 When n=1, 퐷3 there will be one copy of the cycle 퐶3
2 When n=2,퐷3 there will be two copies of the cycle 퐶3
푛 For n-copies of 퐶3 we have the graph 퐷3 where n denotes the number of copies of the cycle 퐶3
Let 푣1 be the common vertex.
Vertex Coloring can be done in the following way:
Assign the color 푐1 to the common vertex 푣1 and the vertices which are connected to the common vertex are assigned with the colors 푐2푐3 … 푐푛
The vertex coloring is done in clockwise direction.
This coloring is a proper vertex coloring which is harmonious (i.e) every pair of color occurs atmost in one pair of adjacent vertices.
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It also follows that only minimum colors should be assigned.
1 1 When n=1 퐷3 , the harmonious chromatic number obtained is 휒퐻(퐷3 )=3
2 2 When n=2 퐷3 , 휒퐻(퐷3 )= 5
3 3 When n=3 퐷3 , 휒퐻(퐷3 )=7
푛 In general, 휒퐻(퐷3 )= 2n+1
푛 Hence , the harmonious chromatic number of Dutch windmill graph is 휒퐻(퐷3 )= 2n+1
Theorem: 3.2
For any n ≥ 2, the harmonious chromatic number of line graph of Dutch windmill graph 풏 is 흌푯[푳(푫ퟑ )]= 2n+1 , where n denotes number of copies of the cycle 퐶3 .
Proof:
푛 Let 퐷3 be the Dutch windmill graph with n-copies of 퐶3
푛 푖 푖 푖 Let V[퐷3 ]={푣1 ,푣2 , 푣3 } be the vertex set where i=1,2,3,….n
푖 1 2 3 푛 Let the vertex 푣1 is the common vertex (i.e) 푣1 = 푣1 = 푣1 = … … = 푣1
푛 푖 푖 푖 Let E[퐷3 ]={푒1 ,푒2 , 푒3 } be the edge set where i=1,2,3,….n
푛 푛 Now construct the line graph of 퐷3 by making every edges of 퐷3 as the vertices 푛 푛 푛 in line graph of 퐷3 . Then the edge set of 퐷3 and the vertex set of 퐿(퐷3 ) are both same.
푛 푖 푖 푖 푛 V[L(퐷3 )]={푢1 ,푢2 , 푢3 } be the newly formed vertex set of line graph of 퐷3 where 푖 푖 푖 푖 푖 푖 푢1 ,푢2 , 푢3 are the newly formed vertices of the edges 푒1 ,푒2 , 푒3 respectively and i=1,2,3,….n
푛 푛 V[L(퐷3 )]=E[퐷3 ]
푖 Now assign the color 푐1 to the common vertex 푣1 where i=1,2,3,….n
푖 푖 The vertices 푣2 , 푣3 where i=1,2,3,…n which are connected to the common vertex are assigned with colors in clockwise direction in such a way that each pair of color occurs exactly once in a pair of adjacent vertices.
푛 Here, for n-copies of 퐶3 in 퐷3 , the minimum number of colors required for harmonious coloring is 2n+1
푛 Hence 휒퐻[퐿(퐷3 )]= 2n+1
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Theorem: 3.3
For n ≥ 3, the harmonious chromatic number of middle graph of the Dutch windmill 풏 graph is 흌푯[푴(푫ퟑ )]= 3n+3
Proof:
푛 Consider the Dutch windmill graph 퐷3 formed by using n-copies of the cycle 퐶3
푛 푖 푖 푖 Let V[퐷3 ]={푣1 ,푣2 , 푣3 } be the vertex set where i=1,2,3,….n
푖 1 2 3 푛 Let the vertex 푣1 is the common vertex (i.e) 푣1 = 푣1 = 푣1 = … … = 푣1
푛 푖 푖 푖 Let E[퐷3 ]={푒1 ,푒2 , 푒3 } be the edge set where i=1,2,3,….n
푛 Construct the middle graph of the Dutch windmill graph 퐷3 and is denoted as 푛 푀(퐷3 ) By the definition of middle graph the vertex set and edge set combines to form the vertex set of the middle graph.
푛 푛 푛 Now, we have, 푉[푀(퐷3 )] = 푉(퐷3 ) ∪ 퐸(퐷3 )
푖 푖 푖 푖 푖 푖 ={푣1 ,푣2 , 푣3 }∪{푒1 ,푒2 , 푒3 } where i=1,2,3,….n
푖 Now assign the color 푐1 to the common vertex 푣1 where i=1,2,3,….n
The edges connected to the common vertex are subdivided and those vertices are considered as inner subdivision vertices. The other vertices are considered as outer subdivision vertices.
The inner subdivision vertices are assigned with the colors 푐2, 푐3, … 푐2푛+1 in clockwise direction.
The outer subdivision vertices are assigned with the colors 푐3푛,푐3푛−1, … 푐3푛+1 in such a way that it is harmonious (i.e) every pair of colors occurs exactly once in pair of adjacent vertices.
3 In 퐷3 and n=3, we can assign the colors 푐2, 푐3, … 푐7 to inner subdivision vertices and the colors 푐8, 푐9, 푐10 to the outer subdivision vertices.
The remaining 2n vertices are assigned with the colors already existing in which no
pair of colors should be repeated where 푐11 = 푐3푛+(푗−1) 푎푛푑 푐12 = 푐3푛+푗 where j=1,2,3,…m
Obviously, this coloring is harmonious in which no pair of colors to adjacent vertices are repeated. Thus the minimum number of colors required to color all the vertices of the 푛 middle graph of 퐷3 is 3n+3
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푛 Hence 휒퐻[푀(퐷3 )]=3n+3, for n≥ 3
Theorem: 3.4
For any n ≥ 2, the harmonious chromatic number of subdivision graph of Dutch 푛 windmill graph is 휒퐻[푆(퐷3 )]=2n+1 where n denotes the number of copies in the cycle 퐶3
Proof:
푛 Let 퐷3 be the Dutch windmill graph with n-copies of 퐶3
푛 푖 푖 푖 Let V[퐷3 ]={푣1 ,푣2 , 푣3 } be the vertex set where i=1,2,3,….n
푖 1 2 3 푛 Let the vertex 푣1 is the common vertex (i.e) 푣1 = 푣1 = 푣1 = … … = 푣1
푛 푖 푖 푖 Let E[퐷3 ]={푒1 ,푒2 , 푒3 } be the edge set where i=1,2,3,….n
푛 Now construct the subdivision graph of 퐷3
By the definition of subdivision graph, the edges are subdivided to form a new vertex in between the two adjacent vertices.
Now the new vertex set of the subdivision graph of Dutch windmill graph is given by
푛 푛 푛 푉[푆(퐷3 )] = 푉(퐷3 ) ∪ 퐸(퐷3 )
푖 푖 푖 푖 푖 푖 ={푣1 ,푣2 , 푣3 }∪{푒1 ,푒2 , 푒3 } where i=1,2,3,….n
푖 푖 푖 푛 The vertices 푒1 ,푒2 , 푒3 are newly formed vertices over the edges of 퐷3
Assign the colors to the vertices in clockwise direction
푖 Assign the color 푐1 to the common vertex 푣1 , i=1,2,3,…n
Next the inner subdivision vertices are assigned with the colors 푐2, 푐3, … 푐2푛+1 in clockwise direction.
The outer subdivision vertices and the remaining vertices which are not adjacent to the common vertex are assigned with the suitable colors.
3 푛 When n=3, in 퐷3 , we have 휒퐻[푆(퐷3 )]=2n+1
3 휒퐻[푆(퐷3 )]=2(3)+1=7 where the coloring in done in the following manner:
Assign the color 푐1 to the common vertex and the colors 푐2, 푐3, … 푐2푛+1
(i.e) the colors 푐2, 푐3, … 푐7 to the inner subdivision vertices
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Then assign the colors 푐2푛−1, 푐2푛, 푐2푛+1 (i.e) 푐5, 푐6, 푐7 to the outer subdivision vertices.
Thus all the vertices in the graph are colored with suitable colors which occurs exactly once in an edge.
Now, the resulting color is harmonious (i.e) proper vertex coloring in which each pair of color occurs at most in one pair of adjacent vertices.
The minimum number of colors required for the harmonious coloring of subdivision graph of Dutch windmill graph is 2n+1
푛 Hence 휒퐻[푆(퐷3 )]=2n+1 , n≥ 2
4. b-coloring
Theorem : 4.1
For any n ≥ 5, the b- chromatic number of the sunlet graph is φ(퐒퐧)=4
Proof:
Let Sn be the sunlet graph. Let V(Sn) = { v1, v2, …. vn}∪{ u1, u2, …. un}.
Now Sn contains 2n vertices where vertices in the cycles taken in cyclic order is denoted by vi where i = 1,2,3,... n
The pendant vertices are denoted by ui where i = 1,2,3,... n such that each ei = viui is ′ ′ ′ a pendant edge where E(Sn) = {e1, e2 ,... en}∪{e1 , e2 , … en }
Let us consider 4 – coloring
Let c1, c2, c3, c4 be the 4 – colors of Sn.
Assign the colors to the vertices using the definition of b-coloring such that it admits proper k-coloring in which every color class has an adjacent vertex to atleast one vertex in each of the other color classes.
Assign the color c1 to v1 , c2 to v2 , c3to v3 , c4to v4 , c3to vn−1 and c4to vn
For pendant vertices, assign c4to u1 and c3to un
Now for 2 ≤ i ≤ n-1, assign the color c1 to ui and for 5 ≤ i ≤ n-2, if there exists, then assign any one of the color which is already existing to that vertex vi because deg(vi)=3
No new colors can be introduced further. This result shows that the coloring is a b- coloring
∴ φ(Sn) ≥ 4 ______(1)
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Using the upper bound property (1.1), φ(G) ≤ ∆(G)+1
∆(Sn)=3. Then we get φ(Sn) ≤ 4 ______(2)
From (1) & (2) , we get,
φ(퐒퐧)=4 for all n ≥ 5.
Theorem : 4.2
If n ≥ 7, then the b-chromatic number of the line graph of sunlet graph is φ[L(퐒퐧)]=5
Proof:
Let Sn be the sunlet graph and let L(Sn) be the line graph of the sunlet graph
In sunlet graph, let v1, v2, …. vn be the vertices of the cycle and let u1, u2, …. un be the pendant vertices such that viui is a pendant edge.
∴ V(Sn)= { v1, v2, …. vn}∪{ u1, u2, …. un}
′ ′ ′ Let e1, e2 ,... en be the edges in the cycle and e1 , e2 , … en be the pendant edges.
′ ′ ′ E(Sn)={e1, e2 ,... en}∪{e1 , e2 , … en }
′ E(Sn)={en} ∪{ ei: 1 ≤ i ≤ n − 1}∪{ei : 1 ≤ i ≤ n}
where en = vnv1
ei = vivi+1 (i=1,2,3,..... n-1)
′ ei = viui (i=1,2,3,..... n)
By the definition, construct the line graph of sunlet graph Sn by making every edges of sunlet graph as vertices in its line graph.
Then the edge set of sunlet graph and vertex set of line graph of sunlet graph are both same.
′ ′ ′ V[L(Sn)]={ui : 1 ≤ i ≤ n}∪{vi : 1 ≤ i ≤ n-1}∪{vn }=E(Sn)
′ ′ ′ where vi and ui are the edges ei and ei respectively.
Consider the 5- coloring and let c1, c2, c3, c4,c5 be the 5 – colors of L(Sn)
′ For 1 ≤ i ≤ 5 , assign the colors ci 푡표 푣푖 where i= 1,2,3,4,5
′ ′ ′ ′ ′ For other 푢푖 assign the colors c1 to 푢5 , c2 to 푢푛−1 , c1 to 푢푛 , c2 to 푢1 ,
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′ ′ ′ c5 to 푢2 , c4 to 푢3 푎푛푑 c5 to 푢4
′ For 8 ≤ i ≤ n assign color c2 to 푢푖
′ ′ For 8 ≤ i ≤ n-1 if any, assign any existing colors to 푣푖 because deg(푣푖 ) = 4
This results in b-coloring.
∴ φ[L(Sn)] ≥ 5 ______(1)
Using the upper bound property (1.1), φ(G)≤∆(G)+1
∆[L(Sn)]=4. Then we get φ[L(Sn)] ≤ 5 ______(2)
From (1) & (2) , we get,
φ[L(퐒퐧)]=5 for all n ≥ 7
Theorem: 4.3
If n ≥ 7 , then the b-chromatic number of middle graph of sunlet graph is φ[M(퐒퐧)]=7
Proof:
Let Sn be the sunlet graph. Let V(Sn) = { v1, v2, …. vn}∪{ u1, u2, …. un}.
Let M(Sn) be the middle graph of sunlet graph
Now Sn contains 2n vertices where vertices in the cycles taken in cyclic order is denoted by vi where i = 1,2,3,... n
The pendant vertices are denoted by ui where i = 1,2,3,... n such that each ei = viui is ′ ′ ′ a pendant edge where E(Sn) = {e1, e2 ,... en}∪{e1 , e2 , … en }
′ E(Sn)={en} ∪{ ei: 1 ≤ i ≤ n − 1}∪{ei : 1 ≤ i ≤ n}
where en = vnv1
ei = vivi+1 (i=1,2,3,..... n-1)
′ ei = viui (i=1,2,3,..... n)
Using the definition of middle graph, construct the middle graph of sunlet graph.
By middle graph definition,
′ ′ V[M(Sn)]={vi: 1 ≤ i ≤ n}∪{ui: 1 ≤ i ≤ n} ∪ {ui : 1 ≤ i ≤ n }∪{vi : 1 ≤ i ≤ n}
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′ ′ ′ where V[M(Sn)]=V(Sn)∪퐸(Sn) and vi and ui are the edges ei and ei of the sunlet graph respectively.
Consider the 7- coloring and let c1, c2, c3, c4,c5, c6, c7 be the 7– colors of M(Sn)
For 1 ≤ i≤ n assign the colors ci to vi and any one color ci to ui because deg(ui)=1
For 7 ≤ i≤ n, if any assign any one existing colors to vi and ui because deg(vi)=3
Now assign the colors ′ ′ ′ ′ ′ ′ ′ c1 to vn−1 , c2 to vn−1 , c3 to v1 , c4 푡표 v2 ,c5 to v3 , c6 to v4 and c7 to v5
′ ′ For 7 ≤ i≤ n-1, if exists then assign any one color to vi because deg( vi ) = 6
This results in b-coloring
∴ φ[M(Sn)] ≥ 7 ______(1)
Using the upper bound property (1.1), φ(G)≤∆(G)+1
∆[M(Sn)]=6. Then we get φ[M(Sn)] ≤ 7 ______(2)
From (1) & (2) , we get,
φ[M(퐒퐧)]=7
Theorem: 4.4
If n ≥ 6, then the b-chromatic number of subdivision graph of sunlet graph is
φ[S(퐒퐧)]=n
Proof:
Let Sn be the sunlet graph and let S(Sn) be the subdivision graph of the sunlet graph
In sunlet graph, let v1, v2, …. vn be the vertices of the cycle and let u1, u2, …. un be the pendant vertices such that viui is a pendant edge.
∴ V(Sn)= { v1, v2, …. vn}∪{ u1, u2, …. un}
By the definition, subdivision graph is the graph obtained from G by replacing each of its edge by a path of length two or equivalently by inserting an additional vertex into each edge of the given graph.
The subdivision graph of sunlet graph is V[S(Sn)]=V(Sn)∪E(Sn)
′ ′ V[S(Sn)]={vi: 1 ≤ i ≤ n}∪{ui: 1 ≤ i ≤ n} ∪ {ui : 1 ≤ i ≤ n }∪{vi : 1 ≤ i ≤ n}
Consider n-coloring of subdivision graph of sunlet graph as b-chromatic
For 1 ≤ i≤ n, assign the color ci to vi since d(vi)=3
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Now assign anyone of the colors to pendant vertex because d(ui)=1
′ ′ Let vi and vi be the vertices newly formed by introducing new vertex in every edge of sunlet graph.
Now assign n-colors to the vertices in clockwise direction.
This results in b-coloring.
∴ φ[S(Sn)] ≥ n ______(1)
Next to prove φ[S(Sn)] ≤ n
Assume that φ[S(Sn)] >n
Therefore, φ[S(Sn)] = n+1 for all n≥6 there must be atleast n+1 vertices of degree n in subdivision graph of sunlet graph all with distinct colors and each adjacent to vertices of all the other colors.
′ Those vertices are vi (i=1,2,3,....n) since they are with degree atleast n which is a contradiction. Therefore, b-coloring with n+1 colors is impossible.
Thus we have φ[S(Sn)] ≤ n ______(2)
From (1) & (2) , we get,
φ[S(퐒퐧)]=n , for all n ≥ 6
5. CONCLUSION
The harmonious chromatic number of line graph, middle graph and subdivision graph of Dutch Windmill graph is obtained and we also found the b-chromatic number of line graph, middle graph and subdivision graph of sunlet graph.
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