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Total Chromatic Number of Star and Bistargraphs 1D International Journal of Pure and Applied Mathematics Volume 117 No. 21 2017, 699-708 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue ijpam.eu Total Chromatic Number of Star and BistarGraphs 1D. Muthuramakrishnanand 2G. Jayaraman 1Department of Mathematics, National College, Trichy, Tamil Nadu, India. [email protected] 2Research Scholar, National College, Trichy, Tamil Nadu, India. [email protected] Abstract A total coloring of a graph G is an assignment of colors to both the vertices and edges of G, such that no two adjacent or incident vertices and edges of G are assigned the same colors. In this paper, we have discussed 2 the total coloring of 푀 퐾1,푛 , 푇 퐾1,푛 , 퐿 퐾1,푛 and 퐵푛,푛and we obtained the 2 total chromatic number of 푀 퐾1,푛 , 푇 퐾1,푛 , 퐿 퐾1,푛 and 퐵푛,푛. AMS Subject Classification: 05C15 Keywords and Phrases:Middle graph, line graph, total graph, star graph, square graph, total coloring and total chromatic number. 699 International Journal of Pure and Applied Mathematics Special Issue 1. Introduction In this paper, we have chosen finite, simple and undirected graphs. Let퐺 = (푉 퐺 , 퐸 퐺 ) be a graph with the vertex set 푉(퐺) and the edge set E(퐺), respectively. In 1965, the concept of total coloring was introduced by Behzad [1] and in 1967 he [2] came out new ideology that, the total chromatic number of complete graph and complete bi-partite graph. A total coloring of퐺, is a function 푓: 푆 → 퐶, where 푆 = 푉 퐺 ∪ 퐸 퐺 and C is a set of colors to satisfies the given conditions. (i) no two adjacent vertices receive the same colors (ii) notwo adjacent edges vertices receive the same colors (iii) no edges and its end vertices receive the same colors The total chromatic number 휒′′ 퐺 of a graph퐺 is the minimumcardinality k such that 퐺may have a total coloring by k colors.Behzad [1] and Vizing [10] conjectured that for every simple graph G has ∆ 퐺 + 1 ≤ 휒′′ (퐺) ≤ ∆ 퐺 + 2, where ∆ 퐺 the maximum degree of G.This conjecture is called the Total Coloring Conjecture(TCC). Rosenfeld [6] and Vijayaditya [9] verified the TCC, for any graph G with maximum degree ≤ 3 and Kostochka [4] for maximum degree≤ 5. In Borodin[3] verified the total coloring conjecture (TCC)for maximumdegree ≥ 9 in planar graphs. In 1992, Yap and Chew [11] proved that any graph G has a total coloring withat most∆ 퐺 + 2 colors if∆ 퐺 ≥ 푉 퐺 − 5. In recent era, total coloring have been extensively studied in different families of graphs. Mohan et.al [5] given the tight bound of Behzad and Vizing conjecture for Corona product of certain classes of graph.S.K. Vaidya et.al [8] prove that the total chromatic number of middle graph, total graph, shadow graph of cycle and one point union of cycle. S. Sudha et.al [7] prove that the total chromatic number of Sudha grid graphs, gear graph and crown graph. In this present paper, we investigate the total chromatic number of 푀 퐾1,푛 , 푇 퐾1,푛 , 퐿 퐾1,푛 2 and 퐵푛,푛 . 2. Total Chromatic Number of푀 퐾1,푛 Definition 2.1. The middle graph of a graph 퐺, denoted by M(퐺) is define as follows, the vertex set of 푀(퐺) is 푉 퐺 ∪ 퐸 퐺 . Two vertices 푥, 푦 in the vertex set of 푀(퐺) are adjacent in 푀(퐺) in case one of the following condition holds: (i) 푥, 푦 are in 퐸(퐺) and 푥, 푦 is adjacent in 퐺. (ii) 푥 is in 푉 퐺 , 푦 is in 퐸 퐺 and 푥, 푦 are incident in 퐺. Theorem 2.2. Let 푀 퐾1,푛 be the middle graph of star graph. Then 푛 + 2, 푛 푖푠 표푑푑 휒" 푀(퐾 ) = , 1,푛 푛 + 3, 푛 푖푠 푒푣푒푛 700 International Journal of Pure and Applied Mathematics Special Issue Proof: Let 푉 퐾1,푛 = 푣 ∪ 푣1 , 푣2 , 푣3 , . 푣푛 and E 퐾1,푛 = 푒1 , 푒2 , 푒3 , . 푒푛 where {푒푖 = 푣푣푖 : 1 ≤ 푖 ≤ 푛}. Now we construct the middle graph of star, each edge ′ {푒푖 = 푣푣푖 : 1 ≤ 푖 ≤ 푛 } in 퐾1,푛 is subdivided by the vertices {푣푖 ∶ 1 ≤ 푖 ≤ 푛}in ′ ′ ′ ′ 푀(퐾1,푛 ) and the vertices {푣, 푣1, 푣2, 푣3, . , 푣푛 } induce a clique of order 푛 + 1. Let it be 퐾푛+1. In 푀(퐾1,푛 ), the vertex set and the edge setis given by ′ ′ ′ ′ 푉 푀(퐾1,푛 ) = 푣 ∪ 푣1 , 푣2 , 푣3 , . 푣푛 ∪ {푣푖 , 푣2, 푣3, . , 푣푛 } and ′ ′ ′ ′ 퐸 푀(퐾1,푛 ) = 푣푣푖 ∶ 1 ≤ 푖 ≤ 푛 ∪ 푣푖 푣푖 : 1 ≤ 푖 ≤ 푛 ∪ 푣푖 푣푗 ∶ 1 ≤ 푖 ≤ 푛 − 1, 푗>푖, 푖+1≤푗≤푛.Now we define the total coloring 푓, such that푓:푆→퐶 as follows, where푆 = 푉(푀(퐾1,푛 )) ∪ 퐸(푀(퐾1,푛 )) and 퐶 = 1, 2, 3, … . 푛 + 3 . The total coloring is obtained by coloring these vertices and edges as follows. We consider the following two cases: Case(i): when 푛 is odd ′ 푓 푣 = 푛 + 2 푓 푣푖 = 푖 ∶ 1 ≤ 푖 ≤ 푛. 푓 푣푖 = 푛 + 1 ∶ 1 ≤ 푖 ≤ 푛 2푖, 푖푓 2푖 ≢ 0 푚표푑 푛 + 2 푓 푣푣′ = for 1 ≤ 푖 ≤ 푛 푖 푛 + 2, 0푡ℎ푒푟푤푖푠푒 푖 + 푗, 푖푓 (푖 + 푗) ≢ 0 푚표푑 푛 + 2 푓 푣′ 푣′ = 푖 푗 푛 + 2, 표푡ℎ푒푟푤푖푠푒 for 1 ≤ 푖 ≤ 푛 − 1 , 푗 > 푖, 푖 + 1 ≤ 푗 ≤ 푛 ′ 푓 푣푖 푣푖 = 푖 − 1 ∶ 2 ≤ 푖 ≤ 푛 ′ 푓 푣1푣1 = 푛 + 2 Based on the above method of totalcoloring, the graph푀(퐾1,푛 ) is total colored with 푛 + 2colors. Hence the total chromatic number of 푀(퐾1,푛 ), 휒"(푀(퐾1,푛 )) = 푛 + 2. Case (ii): when 푛 is even ′ 푓 푣 = 푛 + 3 푓 푣푖 = 푖 ∶ 1 ≤ 푖 ≤ 푛. 푓 푣푖 = 푛 + 1 ∶ 1 ≤ 푖 ≤ 푛 2푖, 푖푓 2푖 ≢ 0 푚표푑 푛 + 3 푓 푣푣′ = for 1 ≤ 푖 ≤ 푛 푖 푛 + 3, 0푡ℎ푒푟푤푖푠푒 푖 + 푗, 푖푓 (푖 + 푗) ≢ 0 푚표푑 푛 + 3 푓 푣′ 푣′ = 푖 푗 푛 + 3, 표푡ℎ푒푟푤푖푠푒 for 1 ≤ 푖 ≤ 푛 − 1 , 푗 > 푖, 푖 + 1 ≤ 푗 ≤ 푛 ′ 푓 푣푖 푣푖 = 푖 − 1 ∶ 2 ≤ 푖 ≤ 푛 ′ 푓 푣1푣1 = 푛 + 2 Based on the above ruleof total coloring, the graph 푀(퐾1,푛 ) is total colored with 푛 + 3colors.Hence the total chromatic number of푀(퐾1,푛 ), 휒"(푀(퐾1,푛 )) = 푛 + 3. 701 International Journal of Pure and Applied Mathematics Special Issue Illustration 2.3: Consider the middle graph of star Figure 1: Total Coloring of Middle Graphof Star 푀(퐾1,7) for n is Odd Illustration 2.4: Consider the middle graph of star Figure 2: Total Coloring of Middle Graph of Star 푀 퐾1,8 for n is Even 3. Total Chromatic Number of 퐓 퐊ퟏ,퐧 Definition 3.1.The Total graph of a graph 퐺, denoted by 푇(퐺) is define as, the vertex set of 푇(퐺) is 푉 퐺 ∪ 퐸 퐺 . Two vertices 푥, 푦 in the vertex set of 푇(퐺) are adjacent in 푇(퐺) in case one of the following condition holds: (i) 푥, 푦 are in 푉(퐺) and 푥 is adjacent to 푦 in 퐺. (ii) 푥, 푦 are in 퐸(퐺) and 푥, 푦 is adjacent in 퐺. (iii) 푥 is in 푉 퐺 , 푦 is in 퐸 퐺 and 푥, 푦 are incident in 퐺. Theorem 3.2. Let 푇 퐾1,푛 be the total graph of star graph. Then 휒" 푇(퐾1,푛 ) = 2푛 + 1, Proof: Let푉 퐾1,푛 = 푣 ∪ 푣1 , 푣2 , 푣3 , . 푣푛 and E 퐾1,푛 = {푒1 , 푒2 , 푒3 , . 푒푛 }where {푒푖 = 푣푣푖 : 1 ≤ 푖 ≤ 푛}. Now we construct the total graphof star, each edge ′ {푒푖 = 푣푣푖 : 1 ≤ 푖 ≤ 푛 } in 퐾1,푛 is subdivided by the vertices {푣푖 ∶ 1 ≤ 푖 ≤ 푛}in ′ ′ ′ ′ 푇(퐾1,푛 ) and the vertices {푣, 푣1, 푣2, 푣3, . , 푣푛 } induce a clique of order 푛 + 1. Let it be 퐾푛+1. 702 International Journal of Pure and Applied Mathematics Special Issue In 푇(퐾1,푛 ), the vertex set and the edge setis given by 푉 푇(퐾1,푛 ) = 푣 ∪ ′ ′ ′ ′ ′ 푣1 , 푣2 , 푣3 , . 푣푛 ∪ {푣1, 푣2, 푣3, . , 푣푛 } and 퐸 푇(퐾1,푛 ) = 푣푣푖 ∶ 1 ≤ 푖 ≤ 푛 ∪ ′ ′ ′ 푣푣푖 ∶ 1 ≤ 푖 ≤ 푛 ∪ 푣푖 푣푖 : 1 ≤ 푖 ≤ 푛 ∪ 푣푖 푣푗 ∶ 1 ≤ 푖 ≤ 푛 − 1, 푗 > 푖, 푖 + 1≤푗≤푛. Now we define the total coloring 푓, such that푓: 푆 → 퐶 as follows, where 푆 = 푉(푇(퐾1,푛 )) ∪ 퐸(푇(퐾1,푛 )) and 퐶 = 1, 2, 3, … .2푛 + 1 . The total coloring is obtained by coloring these vertices and edges as follows. ′ 푓 푣 = 2푛 + 1 푓 푣푖 = 푖 ∶ 1 ≤ 푖 ≤ 푛. 푓 푣푖 = 2푛 ∶ 1 ≤ 푖 ≤ 푛 2푖, 푖푓 2푖 ≢ 0 푚표푑 2푛 푓 푣푣′ = for 1 ≤ 푖 ≤ 푛 푖 2푛, 0푡ℎ푒푟푤푖푠푒 2푖 − 1, 푖푓 2푖 − 1 ≢ 0 푚표푑 2푛 − 1 푓 푣푣 = for 1 ≤ 푖 ≤ 푛 푖 2푛 − 1, 0푡ℎ푒푟푤푖푠푒 푖 + 푗, 푖푓 (푖 + 푗) ≢ 0 푚표푑 2푛 + 1 푓 푣′ 푣′ = 푖 푗 2푛 + 1, 표푡ℎ푒푟푤푖푠푒 for 1 ≤ 푖 ≤ 푛 − 1 , 푗 > 푖, 푖 + 1 ≤ 푗 ≤ 푛 ′ 푓 푣푖 푣푖 = 푖 − 1 ∶ 2 ≤ 푖 ≤ 푛 ′ 푓 푣1푣1 = 푛 + 2 Based on the above rule of total coloring, the graph 푇(퐾1,푛 ) is total colored with 2푛 + 1colors. Hence the total chromatic number of the total graph of 푇(퐾1,푛 ), 휒"(푇(퐾1,푛 )) = 2푛 + 1. Illustration 3.3: Consider the total graph of Star graph Figure 3: Total Coloring of Total Graph of Star 푇(퐾1,8) 4. Total Chromatic Number of퐿(퐵푛,푛 ) Definition 4.1. Bistar graph is the graph obtained by joining the root vertices of two copies of star 퐾1,푛 Definition 4.2.
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