Annals of Pure and Applied Mathematics Vol. 13, No. 2, 2017, 173-183 Annals of ISSN: 2279-087X (P), 2279-0888(online) Published on 4 April 2017 www.researchmathsci.org DOI: http://dx.doi.org/10.22457/apam.v13n2a3
Adjacent Vertex Distinguishing Total Coloring of Line and Splitting Graph of Some Graphs K.Thirusangu 1 and R.Ezhilarasi 2 1Department of Mathematics, S.I.V.E.T College, Gowrivakkam Chennai - 600 073, India 2Ramanujan Institute for Advanced Study in Mathematics University of Madras, Chennai - 600 005, India. 2Corresponding author. Email: [email protected] ; [email protected] Received 9 March 2017; accepted 28 March 2017 Abstract. In this paper, we prove the existence of the adjacent vertex distinguishing total n coloring of line graph of Fan graph Fn , double star K1, n,n , Friendship graph F and 2n splitting graph of path Pn , cycle Cn , star K1, n and sun let graph S in detail. Keywords: simple graph, adjacent vertex distinguishing total coloring, line graph, splitting graph, path, cycle, star, sun let, fan, double star and friendship graph. AMS Mathematics Subject Classification (2010): 05C15, 05C38, 05C76 1. Introduction Throughout this paper, we consider finite, simple, connected and undirected graph G = (V (G), E(G)) . For every vertex u,v ∈V (G) , the edge connecting two vertices is denoted by uv ∈ E(G) . For all other standard concepts of graph theory, we see [1,2,5]. ∪ For graphs G1 and G2 , we let G1 G2 denotes their union, that is, ∪ ∪ ∪ ∪ V (G1 G2 ) = V (G1) V (G2 ) and E(G1 G2 ) = E(G1) E(G2 ) . If a simple graph G ψ ≥ ∆ + has two adjacent vertices of maximum degree, then avt (G) (G) 2 . Otherwise, if a simple graph G does not have two adjacent vertices of maximum degree, then ψ ∆ + avt (G) = (G) 1. The well-known AVDTC conjecture, made by Zhang et al. [8] χ ≤ ∆ + says that every simple graph G has avt (G) (G) 3 . AVDTC of tensor product of graphs are discussed in litrature [6,7]. (2,1)-total labelling of cactus graphs and adjacent vertex distinguishing edge colouring of cactus graphs are discussed in literature [3,4]. Throughout the paper, we denote the path graph, cycle graph, complete graph, star graph, 2n sun let, fan graph, double star and friendship graph by Pn , Cn , K n , K1, n , S , Fn , n K1, n and F respectively. A proper total coloring of G is an assignment of colors to the vertices and the edges such that no two adjacent vertices and adjacent edges are assigned with the same color, no edge and its end vertices are assigned with the same color and for
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K.Thirusangu and R.Ezhilarasi any pair of adjacent vertices have distinct set of colors. A total k -coloring of the graph G is adjacent vertex distinguishing, if a mapping f :V (G) ∪ E(G) → {1,2, ⋯, k}, k ∈Z+ such that any two adjacent or incident elements in V (G) ∪ E(G) have different colors. The minimum number of colors required to give an adjacent vertex distinguishing total coloring (abbreviated as χ AVDTC) to the graph G is denoted by avt (G) . First, we find the adjacent vertex distinguishing total chromatic number of Fan graph, double star graph, Friendship graph and complete graph.
χ + Proposition 1.1. avt (Fn ) = n 1, n > 3 .
Proof: Fan graph denoted by Fn , is a graph obtained by joining all vertices of the path + Pn to a further vertex called the center vertex v . Thus Fn contains n 1 vertices and − { } 2n 1 edges. Here, we have the vertex and edge set V (Fn ) = v,v1,v2 ,⋯,vn and {( n ) ( n−1 )} ∪ → E[F ] = vv ∪ v v + . Now we define f :V (F ) E(F ) {1,2, ⋯,k} n ∪i=1 i ∪i=1 i i 1 n n given by, ≤ ≤ + For 1 i n, f (vi ) = i and f (v) = n 1 ≤ ≤ − + For 1 i n 1, f (vivi+1 ) = i 2andf (vv 1 ) = n ≤ ≤ − For 2 i n, f (vivn ) = i 1 ∴ χ + avt (Fn ) = n 1, n > 3 .
χ + Proposition 1.2. avt (K1, n,n )) = n 1, for n > 2.
Proof: The double star K1, n,n is obtained from the star K1, n by adding a new pendent edge of the existing n pendent vertices. Here v is the apex vertex. The vertex set and edge set of K1, n,n as follows, n n ∪ ∪ ′ ()∪ ′ V[K1, n,n ] = ∪v vi vi and E[K1, n,n ] = ∪ vv i vivi i=1 i=1 + Then |V (K1, n,n |=) 2n 1 and | E(K1, n,n |=) 2n . We define ∪ → f :V (K1, n,n ) E(K1, n,n ) {1,2, ⋯,k} given by + ′ ′ f (v) = n 1, f (vv n ) = 1, f (vn−1vn−1) = 1andf (vnvn ) = 2 . ≤ ≤ ′ + For 1 i n, f (vi ) = iandf (vi ) = n 1, ≤ ≤ − + For 1 i n 1, f (vv i ) = i 1 ≤ ≤ − ′ + For 1 i n 2 , f (vivi ) = i 2 . ∴ χ + avt (K1, n,n )) = n 1 , for n > 2.
χ n + ≥ Proposition 1.3. avt (F ) = 2n 1, n 2 .
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Adjacent Vertex Distinguishing Total Coloring of Line and Splitting Graph of Some Graphs Proof: The friendship graph F n can be constructed by joining n copies of the cycle graph C3 with a common vertex say v . Thus, the vertex set and edge set are n {( 2n ) ( n )} n + {v,v ,v ,⋯,v } and E[F ] = vv ∪ v − v . Here |V (F |=) 2n 1 and 1 2 2n ∪i=1 i ∪i=1 2i 1 2i | E(F n |=) 3n . Now we define f :V (F n ) ∪ E(F n ) →{1,2, ⋯,k} given by for n ≥ 2 , For 1≤ i ≤ n , − + f (v2i−1) = 2, f (v2i ) = 3, f (vv 2i−1) = 2i 1, f (vv 2i ) = 2i and f (v) = f (v2i−1v2i ) = 2n 1.
∴ χ n + ≥ avt (F ) = 2n 1, n 2 .
Proposition 1.4. The complete graph K n admits AVDTC and n +1, if n ≡ 0 (mod 2) χ (K ) = avt n n + 2, if n ≡ (1 mod 2)
Proof: Let x1, x2 ,⋯xn be the vertices of complete graph K n . We define ∪ → f :V (Kn ) E(Kn ) {1,2, ⋯,k} given by Case-1 When n ≡ 0 (mod 2) For 1 ≤ i, j ≤ n
f (xi ) = i + + + f (xi x j ) = i j, if i j = n 1 For i + j ≠ n +1 i + j , if i + j ≡ 0 (mod 2) 2 f (x x ) ≡ i j n +1 (i + j) (mod n +1), if i + j ≡ (1 mod 2) 2 Case-2 When n ≡ (1 mod 2) For 1 ≤ i, j ≤ n
f (xi ) = i, + + + f (xi x j ) = i j, if i j = n 2 For i + j ≠ n + 2 i + j , if i + j ≡ 0 (mod 2) 2 f (x x ) ≡ i j n + 2 (i + j) (mod n + 2), if i + j ≡ (1 mod 2) 2 n +1, if n ≡ 0 (mod 2) ∴χ (K ) = avt n n + 2, if n ≡ (1 mod 2)
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K.Thirusangu and R.Ezhilarasi We observe that these graphs are admits the adjacent vertex distinguishing total coloring conjecture. Next, we discuss the line graph of these graphs admits AVDTC conjecture.
2. AVDTC of line graph A graph G and its line graph L(G) is a graph such that each vertex of L(G) represents an edge of G and two vertices of L(G) are adjacent iff their corresponding edges are adjacent in G . In this section, the adjacent vertex distinguishing total coloring of line n graph of Fan graph L(Fn ) , double star L(K1, n,n ) and Friendship graph L(F ) are discussed.
Theorem 2.1. The line graph of Fan graph L(Fn ) admits AVDTC and χ + avt L(Fn ) = n 3, for n > 3
Proof: From the proposition (1.1), Let us consider the edges of Fn as ≤ ≤ ≤ ≤ − vv i = xi , for 1 i n and vivi+1 = yi , for 1 i n 1. n n−1 ∪ V[L(Fn )] = ∪xi ∪yi i=1 i=1 n2 + 5n −8 Hence |V (L(F )) |= 2n −1 and | E(L(F )) |= . Note that {x , x ,⋯, x } n n 2 1 2 n forms the complete graph K n with n vertices. Now we define ∪ → f :V (L(Fn )) E(L(Fn )) {1,2, ⋯,k}as follows. For this, first we color a clique in
L(Fn ) using the proposition (1.4). Now the remaining vertices and edges of L(Fn ) are colored as follows.
Figure 1: L(F6 )
Case-1. When n is even. ≤ ≤ − + + For 1 i n 1, f (xi yi ) = n 2 ; f (xi+1 yi ) = n 3 ; f (yi ) = f (xi xi+1) ;
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Adjacent Vertex Distinguishing Total Coloring of Line and Splitting Graph of Some Graphs ≤ ≤ − + For 1 i n 2 , f (yi yi+1) = i 1.
Case-2. When n is odd. ≤ ≤ − + For 1 i n 1, f (xi xi+1) = f (yi ) ; f (xi+1 yi ) = n 3 ; ≤ ≤ − + For 1 i n 2 , f (yi yi+1) = i 1, n n For 1 ≤ i ≤ , f (x y ) = i ; f (x − y − ) = + i . 2 2i 2i 2i 1 2i 1 2 Hence Line graph of fan graph admits AVDTC. ∴ χ + avt (L(Fn )) = n 3. Hence the theorem.
Theorem 2.2. The line graph of double star graph L(K1, n,n ) admits AVDTC and χ + avt L(K1, n,n ) = n 2, for n > 3.
Proof: From the proposition (1.2), Let us consider the edges of K1, n,n as vv i = xi and n v v′ = y for 1≤ i ≤ n . V[L(K )] = { (x ∪ y )} Hence |V (L(K )) |= 2n and i i i 1, n,n ∪i=1 i i 1, n,n n2 + n | E(L(K )) |= . Note that {x , x ,⋯, x } forms the complete graph K with 1, n,n 2 1 2 n n ∪ → n vertices. Now we define f :V (L(K1, n,n )) E(L(K1, n,n )) {1,2, ⋯,k} as follows.
For this, first we color a clique in L(K1, n,n ) using the proposition (1.4). Now the remaining vertices and edges are colored as follows.
Figure 2: L(K1,6,6 )
Case-1. When n is even. ≤ ≤ + + For 1 i n , f (xi yi ) = n 2 ; f (yi ) = n 1.
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K.Thirusangu and R.Ezhilarasi Case-2. When n is odd. n For 1≤ i ≤ n , f (y ) = n + 2 . For 1 ≤ i ≤ , f (x y ) = i . i 2 2i 2i n n For 1 ≤ i ≤ , f (x − y − ) = + i . 2 2i 1 2i 1 2 Hence the Line graph of double star admits AVDTC. ∴ χ + avt (L(K1, n,n )) = n 2. Hence the theorem.
Theorem 2.3. The line graph of friendship graph L(F n ) admits AVDTC and χ n + ≥ avt L(F ) = 2n 3, for n 2 Proof: From the proposition (1.3), Let us consider the edges of F n as ≤ ≤ ≤ ≤ v2i−1v2i = yi , for 1 i 2n and vv i = xi , for 1 i n . n 2n n ∪ V[L(F )] = ∪yi ∪xi i=1 i=1
Figure 3: L(F 3 ) Hence |V (L(F n )) |= 3n and | E(L(F n )) |= n(2 n +1) . Note that
{x1, x2 ,⋯, x2n} forms the complete graph K2n with 2n vertices and x2i−1 and x2i are ≤ ≤ adjacent with yi for 1 i n . Now we define f :V (L(F n )) ∪ E(L(F n )) →{1,2, ⋯, k} as follows. For this, first we color a clique in L(F n ) using the poposition (1.4). The remaining are colored for 1≤ i ≤ n , + + f (x2i−1 yi ) = n 2 ; f (x2i xi ) = n 3 ; f (yi ) = f (x2i−1x2i ) . Hence the Line graph of friendship graph admits AVDTC. ∴ χ (L(F n )) = 2n + 3. avt
3. AVDTC of splitting graph For a graph G the splitting graph of G is obtained by adding a new vertex v′
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Adjacent Vertex Distinguishing Total Coloring of Line and Splitting Graph of Some Graphs corresponding to each vertex v of G such that N (v) = N (v′) . The resultant graph is denoted as Sp (G) . In this section, the adjacent vertex distinguishing total coloring of 2n Splitting graph of path Pn , cycle Cn and star K1, n and sun let S graphs are discussed.
Theorem 3.1. The splitting graph of path graph Sp (Pn ) admits AVDTC and 6, for n ≥ 4 χ Sp (P ) = avt n 5, for n = 3 ′ ′ ′ Proof: Let we denote {v1,v2 ,⋯,vn} as the vertices of Pn and {v1,v2 ,⋯,vn} as the new vertices. The vertex set and edge set of Sp (Pn ) are n n−1 ∪ ′ ()∪ ′ ∪ ′ V[Sp (Pn )] = ∪vi vi and E[Sp (Pn )] = ∪ vivi+1 vivi+1 vivi+1 i=1 i=1
Figure 4: Sp (P5 ) − Hence |V (Sp (Pn )) |= 2n and | E(Sp (Pn )) |= 3( n 1) . This theorem is trivial for n = 3 . ∪ → ≥ Now we define f :V (Sp (Pn )) E(Sp (Pn )) {1,2, ⋯,k} given by for n 4 , {1}, if i ≡ (1 mod 2) For 1 ≤ i ≤ n, f (v ) = f (v′) = i i {2}, if i ≡ 0 (mod 2) {3}, if i ≡ (1 mod 2) For 1 ≤ i ≤ n −1, f (v v + ) = i i 1 {4}, if i ≡ 0 (mod 2) ′ ′ f (vivi+1) = 5 and f (vivi+1) = 6 Clearly, the color classes of any two adjacent vertices are different. ∴ χ ≥ avt (Sp (Pn )) = 6, n 4. Hence the theorem.
Theorem 3.2. The splitting graph of cycle graph Sp (Cn ) admits AVDTC and χ ≥ avt Sp (Cn ) = 6, n 4. ′ ′ ′ Proof: Let we denote {v1,v2 ,⋯,vn} as the vertices of Cn and {v1,v2 ,⋯,vn} as the new vertices. The vertex set and edge set of Sp (Cn ) is given by
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K.Thirusangu and R.Ezhilarasi n ∪ ′ V[Sp (Cn )] = ∪vi vi and i=1
n−1 ()∪ ′ ∪ ′ ∪ ∪ ′ ∪ ′ E[Sp (Cn )] = ∪ vi vi+1 vi vi+1 vi vi+1 (vn v1 vnv1 vn v1 ) i=1
Figure 5: Sp (C6 )
Then |V (Sp (Cn )) |= 2n and | E(Sp (Cn )) |= 3n . ∪ → Define f :V (Sp (Cn )) E(Sp (Cn )) {1,2, ⋯, k} as follows. Case-1. When n is even. 1, if i ≡ (1 mod 2) For 1≤ i ≤ n, f (v ) = f (v′) = i i 2, if i ≡ 0 (mod 2)
3, if i ≡ (1 mod 2) ≤ ≤ − For 1 i n 1, f (vivi+1) = 4, if i ≡ 0 (mod 2) ′ ′ ′ ′ f (vivi+1) = f (vnv1) = 6, f (vivi+1) = f (vnv1) = 5, andf (vnv1) = 4. Case-2. When n is odd.
1, if i ≡ (1 mod 2) ≤ ≤ − ′ For 1 i n 1, f (vi ) = f (vi ) = 2, if i ≡ 0 (mod 2)
3, if i ≡ (1 mod 2) ≤ ≤ − For 1 i n 2, f (vivi+1) = 4, if i ≡ 0 (mod 2)
′ ′ ′ ′ For 1≤ i ≤ n −1, f (v v + ) = f (v v ) = 6, f (v v + ) = f (v v ) = 5, f (v ) = 4, i i 1 n 1 i i 1 n 1 n f (vnv1) = 2 and f (vn−1vn ) = 1 The color classes of any two adjacent vertices are different.
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Adjacent Vertex Distinguishing Total Coloring of Line and Splitting Graph of Some Graphs ∴ χ avt (Sp (Cn )) = 6 . Hence the theorem.
Theorem 3.3. The splitting graph of star graph Sp (K1, n ) admits AVDTC and χ + ≥ avt Sp (K1, n ) = 2n 1, n 2. ′ ′ ′ ′ Proof: Let {v,v1,v2 ,⋯,vn} be the vertices of star K1, n and {v ,v1,v2 ,⋯,vn} be the new vertices of K1, n . Here v is the apex vertex. The vertex set and edge set of Sp (K1, n ) is given by
Figure 6: Sp (K1,4 ) n n ∪ ′ ∪ ∪ ′ ()∪ ′ ∪ ′ V[S(K1, n )] = ∪vi vi v v and E[Sp (K1, n )] = ∪ vv i vv i v vi i=1 i=1 + Then |V (Sp (K1, n )) |= 2( n 1) and | E(Sp (K1, n )) |= 3n . ∪ → We define f :V (Sp (K1, n )) E(Sp (K1, n )) {1,2, ⋯,k} given by For 1≤ i ≤ n , f (v ) = f (v′) = i, f (v) = f (v′) = n +1, f ( vv ′) = f (v′v ) = n +1+ i i i i i
For 1≤ i ≤ n −1, + f (vv i ) = i 1, f (vv n ) = 1 Clearly, the color classes of any two adjacent vertices are different. ∴ χ + ≥ avt (Sp (K1, n )) = 2n 1, n 2. Hence the theorem.
Theorem 3.4. The splitting graph of sun let graph S 2n admits AVDTC and χ 2n ≥ avt Sp (S ) = 8, n 4. ′ ′ ′ 2n Proof: Let {v1,v2 ,⋯,vn ,v1,v2 ,⋯vn} be the vertices of sun let graph S . Here ′ deg (vi ) = 3 and deg (vi ) = 1 for i = 1,2, ⋯,n . Now consider the new vertices ′ ′ ′ 2n 2n {u1,u2 ,⋯,un ,u1,u2 ,⋯,un} of S . The vertex set and edge set of Sp (S ) is given by
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K.Thirusangu and R.Ezhilarasi n 2n ∪ ′ ∪ ∪ ′ V[Sp (S )] = ∪vi vi ui ui i=1 n−1 n 2n ()()()∪ ∪ ∪ ′ ∪ ′ ∪ ′ ∪ ∪ ∪ E[Sp (S )] = ∪ vivi+1 viui+1 vi+1ui ∪ vivi viui uivi vnv1 vnu1 v1un i=1 i=1 Then |V (Sp (S 2n )) |= 4n and | E(Sp (S 2n )) |= 6n .
Figure 7: Sp (S 6 ) We define f :V (Sp (S 2n )) ∪ E(Sp (S 2n )) →{1,2, ⋯, k} given by Case-1. If n ≡ (1 mod 2) . For 1≤ i ≤ n −1 1, if i ≡ (1 mod 2) f (v ) = f (u ) = i i 2, if i ≡ 0 (mod 2) For 1≤ i ≤ n − 2 3, if i ≡ (1 mod 2) f (v v + ) = i i 1 4, if i ≡ 0 (mod 2) and f (vn ) = 4 , f (vnv1) = 2 , f (vn−1vn ) = 1 For 1≤ i ≤ n −1
f (viui+1) = f (vnu1) = 5, f (vi+1ui ) = f (v1un ) = 6 For 1≤ i ≤ n ′ ′ ′ ′ ′ f (vi ) = f (ui ) = 3, f (vivi ) = 7 and f (uivi ) = f (viui ) = 8 Case-2. If n ≡ 0 (mod 2) . For 1≤ i ≤ n 1, if i ≡ (1 mod 2) f (v ) = f (u ) = i i 2, if i ≡ 0 (mod 2) For 1 ≤ i ≤ n −1 3, if i ≡ (1 mod 2) f (v v + ) = and f (v v ) = 4. i i 1 4, if i ≡ 0 (mod 2) n 1
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Adjacent Vertex Distinguishing Total Coloring of Line and Splitting Graph of Some Graphs ≤ ≤ − For 1 i n 1, f (viui+1) = f (vnu1) = 5, f (vi+1ui ) = f (v1un ) = 6 ≤ ≤ ′ ′ ′ ′ ′ For 1 i n, f (vi ) = f (ui ) = 3, f (vivi ) = 7 and f (uivi ) = f (viui ) = 8 ∴ χ 2n ≥ avt (Sp (S )) = 8, n 4. Therefore the splitting graph of sun let graph admits AVDTC conjecture. Hence the theorem.
4. Conclusion We found the adjacent vertex distinguishing total chromatic number of line graph of Fan graph, double star, Friendship graph and splitting graph of path, cycle, star and sun let graph. Also, it is proved that the adjacent vertex distinguishing total coloring conjecture is true for these graphs.
Acknowledgement. The second author gratefully acknowledge The Department of Science and Technology, DST PURSE – Phase II for the financial support under the grant 686 (2014). REFERENCES 1. N.L.Biggs, Algebraic Graph Theory, Cambridge University Press, 2nd edition, Cambridge, 1993. 2. F.Harary, Graph Theory, Narosa publishing House. (2001). 3. N.Khan, M.Pal and A.Pal, (2,1)-total labelling of cactus graphs, Journal of Information and Computing Science, 5(4) (2010) 243-260. 4. N.Khan and M.Pal, Adjacent vertex distinguishing edge colouring of cactus graphs, International Journal of Engineering and Innovative Technology , 4(3) (2013) 62-71. 5. K.K.Parthasarathy, Basic Graph Theory, Tata McGraw Hill, (1994). 6. K.Thirusangu and R.Ezhilarasi, Adjacent-vertex- distinguishing total coloring of tensor product of graphs, communicated. 7. K.Thirusangu and R.Ezhilarasi, Adjacent vertex distinguishing total coloring of quadrilateral snake, Journal of Advances in Mathematics, 13 (1) (2017) 7030-7041. 8. Z.Zhang, X.Chen, L.Jingwen, B.Yao, X.Lu and J.Wang, On adjacent-vertex- distinguishing total coloring of graphs, Science in China Ser. A Mathematics, 48 (3) (2005) 289-299.
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