International Journal of Pure and Applied Mathematics Volume 119 No. 15 2018, 1113-1123 ISSN: 1314-3395 (on-line version) url: http://www.acadpubl.eu/hub/ Special Issue http://www.acadpubl.eu/hub/

on b-coloring of mycielskian graphs

M.Kalpana,Ph.D Research scholar, Kongunadu arts and science college, Coimbatore_29, Tamil nadu, India. D.Vijayalakshmi, Assistant professor, Kongunadu arts and science college, Coimbatore_29, Tamil nadu,

India.

Abstract The b-chromatic number of G, denoted by φ(G),is the maximum k for which G has a b-coloring by k colors. A b-coloring of G by k colors is a proper k-coloring of the vertices of G such that in each color class i there exists a vertex xi having neighbors in all the other k-1 color classes. Such a vertex xi is called a b-dominating vertex, and the set of vertices {x1,x2,…xk}is called a b-dominating system. In this paper, we investigate the b-chromatic number of Mycielski’s graph of path graph, cycle graph, complete graph, wheel graph, gear graph, helm graph denoted by µ(Pn), µ(Cn), µ(Kn), µ(Wn), µ(Gn) and µ(Hn) respectively. Keywords--- b-coloring, b-chromatic number, Mycielskian graph.

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1. Introduction

Graph theory is the theory of graphs dealing with nodes and connections or vertices and edges. The subject has experienced explosive growth, due in large measure to its role as an essential structure underpinning modern applied mathematics. Configurations of nodes and connections has great diversity of applications. They may represent physical networks, such as electrical circuits, roadways, or organic molecules. They are also used in representing less tangiable interactions as might occur in ecosystem, sociological relationships, database, or in the flow of control in a computer program. It is a fastest growing field in Mathematics mainly because of its applications in distinct areas. There are plenty of works have been done in different topics, like decomposition, domination, factoring, orienting, coloring etc, in the past decades. In Graph theory, a graph G = (V,E), consists of two sets V and E. The elements of V are called vertices or nodes. The elements of E are called edges or connections. In this paper, all the graphs considered as loop less, undirected and having no multiple edges. Graph coloring deals with the general and widely applicable concept of partitioning the underlying set of a structure into parts, each of which satisfies a given requirement. Here coloring means a vertex coloring of a graph. A k-coloring of a graph G = (V,E)is a mapping C : V → S where S is a set of k colors; thus k-coloring is an assignment of k colors to the vertices of G. Usually, the set S of colors is taken to be {1,2,...k}.

A coloring C is proper if no two adjacent vertices are assigned the same color. Only loop less graphs admits proper coloring.

The minimum k for which a graph G is k-colorable is called its chromatic number, and denoted χ(G). If χ(G) = k, the graph G is said to be k chromatic. From past few decades, a large number of researchers have been showed their interest to obtain different chromatic numbers for the Mycielski’s Graphs. In this paper we investigate the b-chromatic number of graphs obtained by many different Graph Constructions from a Mycielski’s graph.

The b-chromatic number of G, denoted by φ(G), is the maximum k for which G has a b-coloring by k colors. A b-coloring of G by k colors is a proper k- coloring of the vertices of G such that in each color class i there exists a vertex xi having neighbors in all the other k-1 color classes. Such a vertex xi is called a b-dominating vertex, and the set of vertices {x1,x2 ...xk} is called a b-dominating system. The b-coloring was introduced by R.W.Irving and D.F.Manlove in [7]. They proved that determining φ(G) is NP-hard in general and polynomial for trees.

In a search for triangle-free graphs with arbitrary large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph µ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G) + 1. The Mycielski

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construction is a method for turning a triangle free graph with chromatic number k into larger triangle free graph with chromatic number k+1. It works as follows: As input, take a triangle free graph G with χ(G) = k and the vertex set {v1,v2,...vn}. From the Graph G′ as follows: Let V (G′) = {v1,v2,...vn}∪{u1,u2,...un}∪{w}. Start with E (G′) = E (G). For every ui, add edges from ui to all of vi’s neighbors. Finally, attach an edge from w to every vertex {u1,u2,...un}. This paper investigate the b-chromatic number of Mycielski’s graphs. In particular, the following results are proved in this paper. Mycielski’s graph of path graph, cycle graph, complete graph, wheel graph, gear graph, helm graph denoted by µ(Pn), µ(Cn), µ(Kn), µ(Wn), µ(Gn) and µ(Hn) respectively. 2. b-Coloring of Mycielski’s Graph of Path Graph Theorem 2.1

For a path graph Pn, n ≥ 5, the b-chromatic number of Mycielski’s graph of path graph is ∆(Pn) + 2. i.e.,φ[µ(Pn)] = ∆(Pn) + 2. Proof

In graph theory, a path in a graph is a finite or infinite sequence of edges which connect a sequence of vertices and all are distinct from one another. Let V (Pn) = {vi : 1 ≤ i ≤ n} be the set of vertices of Pn. By the construction of Mycielski’s Graph, V(µ(Pn)) = V(Pn)∪{ui : 1 ≤ i ≤ n}∪{w} and E(µ(Pn)) = E(Pn)∪{uiv:v ϵ NPn(vi)∪{w},i = 1,2...n}. Let C = {c1,c2,c3,c4} be the corresponding colors. Assign the following coloring for µ(Pn) as b-chromatic:

Case 1: For {vi : 1 ≤ i ≤ n} assign the colors c4,c2,c3 alternatively.

Case 2: For {ui : 1 ≤ i ≤ n} assign the same colors of its corresponding copy of vi.

Case 3: For {w} assign the color c1. Therefore φ[µ(Pn)] ≤ 4.

To prove φ[µ(Pn)] ≥ 4. Let us suppose that φ[µ(Pn)] is less than 4 say φ[µ(Pn)] = 3. In µ(Pn) we have three vertices having degree four and three vertices having degree three, so that we can assign one more color which is also having neighbors of other colors.

This gives the contradiction to φ[µ(Pn)] = 3.Thus φ[µ(Pn)] ≥ 4. Hence φ[µ(Pn)] = 4 = ∆(Pn) + 2.This completes the proof of the theorem. Remark

For a path graph Pn, n ≤4, φ [μ (Pn)] = Δ(Pn) + 1.

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3. b- Coloring of Mycielski’s Graph of Cycle Graph Theorem 3.1

For a cycle graph Cn, n≥5, the b- chromatic number of Mycielski’s graph of cycle graph is Δ(Cn) + 2.

i.e., φ [μ (Cn)] = Δ(Cn) + 2. Proof

In graph theory, a cycle in a graph is closed walk consists of a sequence of vertices starting and ending at the same vertex, with each two consecutive vertices in the sequence adjacent to each other in the graph.

Let V(Cn) = {vi: 1≤ i ≤ n} be the set of vertices of Cn taken in cyclic order. By the construction of Mycielski’s Graph,

V (μ(Cn)) = V(Cn) U{ui: 1 ≤ i ≤ n}U {w} and

E(μ (Cn)) = E (Cn)U{uiv : v ϵ NCn(vi) U{w} , i = 1, 2. . . ..n}.

Let C = {c1, c2, c3, c4} be the corresponding colors.

Assign the following coloring for (μ (Cn)) as b-chromatic:

Case 1 : For {vi: 1≤ i ≤ n} assign the colors c2, c3, c4 alternatively.

Case 2 : For {ui: 1 ≤ i ≤ n} assign the same colors of its corresponding copy of vi.

Case 3 : For {w}-assign the color c1.

Therefore φ [μ (Cn)] ≤4. To prove φ [μ (Cn)] ≥4. Let us suppose that φ [μ (Cn)] is less than 4 say φ [μ(Cn)] = 3.

In μ (Cn) the outer cycle having the degree four and inner cycle having the degree three, so that we can assign one more color which is also having neighbors of other colors.

This gives the contradiction to φ [μ (Cn)] = 3.

Thus φ [μ(Cn)] ≥4. Hence φ [μ(Cn)] = 4 = Δ(Cn) + 2.

This completes the proof of the theorem. Remark

For a Cycle graph Cn, φ[µ(Cn)] = n + 1 for n = 2,3.

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φ[µ(Cn)] = ∆(Cn) + 1 for n = 4. 4. b-Coloring of Mycielski’s Graph of Complete Graph Theorem 4.1

For a complete graph Kn,n ≥ 2, the b-chromatic number of Mycielskis graph of complete graph is n + 1. i.e.,φ[µ(Kn)] = n + 1 Proof

In graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge.

Let V (Kn) = {vi : 1 ≤ i ≤ n} be the set of vertices of Kn. By the construction of Mycielskis Graph,

V(µ(Kn)) =V (Kn) ∪ {ui : 1 ≤ i ≤ n}∪{w} and

E(µ(Kn)) = E (Kn)∪{uiv : v ∈ NKn (ui)∪{w},i = 1,2...n}. Since w is adjacent to each vertex of {ui : 1 ≤ i ≤ n}, also µ(Kn) contains an n clique, φ[µ(Kn)] ≥ n and hence

φ[µ(Kn)] ≥ n. Let C = {c1,...,cn+1} be the corresponding colors. Assign the following coloring for µ(Kn) as b-chromatic:

Case 1: For {vi : 1 ≤ i ≤ n} assign the color ci+1, i = 1,2,...,n + 1.

Case 2: For {ui : 1 ≤ i ≤ n} assign the same colors of its corresponding copy of vi.

Case 3: For {w}-assign the color c1. Therefore φ[µ(Kn)] ≤ n + 1. To prove φ[µ(Kn)] ≥ n + 1.

Let us suppose that φ[µ(Kn)] is less than n+1 say φ[µ(Kn)] = n. Since v1,v2, ...vn in Kn receives distinct colors c1,c2, ...cn, respectively. The same colors can be used to color the vertices u1,u2, ....un. Since w is adjacent with each of u1,u2 ....un, there exists a new color to color w. Which is a contradiction to φ[µ(Kn)] = n. Thus φ[µ(Kn)] ≥ n + 1.Hence φ[µ(Kn)] = n + 1. This completes the proof of the theorem.

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5. b-Coloring of Mycielski’s Graph of Wheel Graph Theorem 5.1

For a wheel graph Wn, n ≥ 6, the b-chromatic number of Mycielskis graph of wheel graph is δ(Wn) + 2. i.e.,φ[µ(Wn)] = δ(Wn) + 2. Proof

In graph theory, a wheel in a graph is a graph formed by connecting a single vertex to all vertices of a cycle. The wheel graph has n vertices and 2(n−1) edges. Let V(Wn) = {vi : 1 ≤ i ≤ n} and E(Wn) = {vnvi : 1 ≤ i ≤ n}∪{vivi+1 : 1 ≤ i ≤ n−1}∪{vn-1v1}. By the Mycielski’s construction, V(µ(Wn)) = V(Wn)∪{ui : 1 ≤ i ≤ n}∪{z}. In µ(Wn), each ui (1 ≤ i ≤ n) is adjacent with each vertex of NWn (vi), and z is adjacent with each vertex of {ui : 1 ≤ i ≤ n}. By the definition of Mycielskian, vn is adjacent with each of {vi : 1 ≤ i ≤ n}∪{ui : 1 ≤ i ≤ n}. The color which is assigned to vn cannot be assigned to any vertex of {vi : 1 ≤ i ≤ n}∪{ui : 1 ≤ i ≤ n} and hence it can be assigned at most two times (possibly to z) in µ(Wn).

Assign the following coloring for µ(Wn) as b- chromatic:

Let C = {c1,...,c5} be the corresponding colors used to color the vertices of µ(Wn).

(i) For {vi : 1 ≤ i ≤ n}

if i = n assign the color c5.

if {vi : 1 ≤ i ≤ n−1} assign the colors c2,c3,c4 alternatively. (ii) For {ui : 1 ≤ i ≤ n}

if i = n assign the color c1.

if {ui : 1 ≤ i ≤ n−1} assign the same colors which is used to color the vertices of ui.

(iii) For {z} assign the color c5.

Therefore, φ[µ(Wn)] ≤ 5. Let us assume that φ[µ(Wn)] is greater than five, say φ[µ(Wn)] = 6. But if we assign the sixth color to the graph µ(Wn), it does not satisfies the condition of b- coloring, that is each color class contains a vertex that has a neighbor in all other color classes. This is a contradiction. Therefore b- coloring with six colors is impossible.

Thus, we have φ[µ(Wn)] ≥ 5.

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Hence, φ[µ(Wn)] = 5 = δ(Wn) + 2.Hence the proof. 6. b- Coloring of Mycielski’s Graph of Helm Graph Theorem 6.1

For a Helm graph Hn, n ≥ 7, the b-chromatic number of Mycielskis graph of helm graph is δ(Hn) + 5.

i.e.,φ[µ(Hn)] = δ(Hn) + 5.

Proof

In graph theory, a helm graph is a graph obtained from an n-wheel graph by adjoining a pendent edge at each node of the cycle.

Let V(Hn) = {vi : 1 ≤ i ≤ n}∪{ui : 1 ≤ i ≤ n} and

E(Hn)={vnvi:1≤i≤n}∪{vivi+1 : 1 ≤ i ≤ n-2}∪{vn-1v1}∪{viui : 1 ≤ i ≤n}. By the ʹ ʹ Mycielskis construction, V(µ(Hn)) = V (Hn)∪{vi : 1 ≤ i ≤ n}∪{ui : 1 ≤ i ≤ ʹ ʹ n}∪{w}. In µ(Hn),vi is adjacent with each of{vi : 1 ≤ i ≤ n}, each vi is adjacent ʹ with the neighbor of vi(1 ≤ i ≤ n), w is adjacent with each of{vi : 1 ≤ i ≤ ʹ ʹ n}and{ui : 1 ≤ i ≤ n}. Also each ui is adjacent with {vi: 1 ≤ i ≤ n}. Let C = {c1, ...,c6} be the corresponding colors used to color the vertices ofµ(Hn).

Assign the following coloring for µ(Hn) as b- chromatic:

Case 1: For {vi : 1 ≤ i ≤ n}

Sub case (i): For {vi : 1 ≤ i ≤ 2} assign the color ci+1.

Sub case (ii): For {vi : 3 ≤ i ≤ n-2} assign the colors c5,c4 alternatively.

Sub case (iii): For {vi : n-1 ≤ i ≤ n} assign the colors c2,c1 respectively.

Case 2: For {ui : 1 ≤ i ≤ n-1}

Sub case (i) : For {ui : 1 ≤ i ≤ n-6}, assign the colors c5,c4,c3,c2,c1 respectively.

Sub case (ii) : For {ui : n-6 ≤ i ≤ n-1}, assign the color c6.

ʹ Case 3: For {vi : 1 ≤ i ≤ n}

Sub case (i) : For v1ʹ and vnʹ assign the colors c4 and c6.

ʹ Sub case (ii) : For {vi : 2 ≤ i ≤ 3} assign the colors ci+2.

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ʹ Sub case (iii) : For {vi : 4 ≤ i ≤ 7} assign the colors ci-2.

ʹ Sub case (iv) : For {vi : 8 ≤ i ≤ n} assign the color c5.

ʹ Case 4: For {ui : 1 ≤ i ≤ n-1} assign the color c6.

Case 5 : For {w} assign the color c1.

Therefore φ[µ(Hn)] ≤ 6. To prove φ[µ(Hn)] ≥ 6. Let us suppose that φ [µ(Hn)] is greater than six, say φ[µ(Hn)] = 7. But if we assign the seventh color to the graph µ(Hn), it does not satisfies the condition of b- coloring, that is each color class contains a vertex that has a neighbor in all other color classes. This is a contradiction. Therefore b-coloring with seven colors is impossible. Thus, we have φ[µ(Hn)] ≥ 6.

Hence, φ[µ(Hn)] = 6 = δ(Hn) + 5.Hence the proof. 7. b-Coloring of Mycielski’s Graph of Gear Graph Theorem 7.1

For a gear graph Gn, n ≥ 4, the b-chromatic number of Mycielski’s graph of gear graph is δ(Gn) + 3.

i.e.,φ[µ(Gn)] = δ(Gn) + 3. Proof

In graph theory, the gear graph Gn, also known as a bipartite wheel graph, is a wheel graph Wn with a graph vertex added between each pair of adjacent graph vertices of the outer cycle. Let V(Gn) = {v0}∪{vi : 1 ≤ i ≤ n}∪{ui : 1 ≤ i ≤ n} and E(Gn) = {v0vi : 1 ≤ i ≤ n}∪{viui : 1 ≤ i ≤ n}∪{uivi+1 : 1 ≤ i ≤ n-1}∪ {unv1}. ʹ ʹ By the Mycielski’s construction, V(µ(Gn)) = V (Gn)∪{vi : 1 ≤ i ≤ n}∪{ui : 1 ≤ i ≤ n}∪{w}.In µ(Gn),

ʹ ʹ v0 is adjacent with each vertex of {vi : 1 ≤ i ≤ n}, each vi is adjacent with the ʹ neighbor of vi (1 ≤ i ≤ n) and each ui is adjacent with the neighbors of ui (1 ≤ i ≤ ʹ ʹ ʹ n), w is adjacent with each of {vi : 1 ≤ i ≤ n} and {ui : 1 ≤ i ≤ n}∪{v0 }. Also ʹ each ui is adjacent with {vi : 1 ≤ i ≤ n}.Let C = {c1,...,c5} be the corresponding colors used to color the vertices ofµ(Gn).

Assign the following coloring for µ(Gn) as b- chromatic:

Case 1: For {vi : 0 ≤ i ≤ n}

Sub case (i) : For {vi : 0 ≤ i ≤ 3} assign the color ci+2.

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Sub case (ii) : For {vi : 4 ≤ i ≤ n} assign the color c5.

Case 2: For {ui : 1 ≤ i ≤ n}

Sub case (i) : For u1 assign the color c5.

Sub case (ii) : For {ui : 2 ≤ i ≤ 3}, assign the color ci+1.

Sub case (iii) : For {ui : 4 ≤ i ≤ n} assign the color c4.

ʹ Case 3: For {vi : 0 ≤ i ≤ n}

ʹ Sub case (i) : For {vi : 0 ≤ i ≤ n-1} assign the colors ci+2.

ʹ ʹ Sub case (ii) : For vn and u1 assign the color c5.

ʹ Case 4: For {ui : 1 ≤ i ≤ n-1}

ʹ Sub case (i) : For {ui : 2 ≤ i ≤ 3} assign the colors ci+1.

ʹ Sub case (ii) : For {ui : 4 ≤ i ≤ n} assign the color c4.

Case 5: For {w} assign the color c1.

Therefore φ[µ(Gn)] ≤ 5. To prove φ[µ(Gn)] ≥ 5. Let us suppose that φ [µ(Gn)] is greater than five, say φ[µ(Gn)] = 6. But if we assign the sixth color to the graph µ(Gn), it does not satisfies the condition of b- coloring, that is each color class contains a vertex that has a neighbor in all other color classes. This is a contradiction. Therefore b-coloring with six colors is impossible. Thus, we have φ[µ(Gn)] ≥ 5.

Hence, φ[µ(Gn)] = 5 = δ(Gn) + 2.Hence the proof. References [1] M. Behzad, “A criterion for the planarity of the total graphof a graph”, Proc. Cambridge Philos.soc., 63, 679-681, (1967). [2] J.A.Bondy and U.S.R. Murty, “Graph Theory with Applications”, MacMillan, London, (1976). [3] B. Effantin, “The b-chromatic number of power graphs ofcomplete caterpillars”, J. Discrete Math. Sci. Cryptogr. 8, 483–502, (2005). [4] B. Effantin - H. Kheddouci, “The b-chromatic number ofsome power graphs”, Discrete Math. Theor. Comput. Sci. 6,45–54, (2003). [5] A. Gallian, “A dynamic survey of Graph Labeling”, (2009).

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[6] T.Hamada and I.Yoshimura, “Traversability and connecting of middle graph of a graph”, Dis.Math, 14, 247-255, (1976). [7] R. W. Irving - D. F. Manlove, “The b-chromatic number of a graph”, Discrete Appl.Math,91, 127–141, (1999). [8] S. Klavzar - M. Jakovac, “The b-chromatic number of cubic graphs”, Preprint series 47 (2009), 1067, January 27, (2009). [9] J.Mycielski, “Sarlecoloriagedes graphes”, Coll.Math. 3,161-162, (1955) [10] V. J. Vernold - M. Venkatachalam - M. M. Akbar Ali, A“Note on Achromatic Coloring of Star Graph Families”, Filomat, 23, 251–255, (3)(2009).

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