CHAPTER 5. b-colouring of Line Graph and Line Graph of Central Graph

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1 CHAPTER 5 b-colouring of Line Graph and Line Graph of Central Graph In this Chapter, the b-chromatic number of L(K 1,n ), L(C n ), L(P n ), L(K m,n ), L(K 1,n,n ), L(F 2,k ), L(B n,n ), L(P m ӨS n ), L[C(K n )], L[C(C n )], L[C(K 1,n )] are obtained along with its structural properties. 5.1 Introduction [32, 33, 38, 46, 76, 81] In Graph theory, the Line graph L(G) of undirected graph G is another graph L(G) that represents the adjacencies between the edges of G. Other terms used for the Line graph are the covering graph, the edge-to-vertex dual, the conjugate, the representative graph, the edge graph, the interchange graph, the adjoint graph and the derived graph. One of the earliest and most important theorems about Line graphs is due to Hassler Whitney (1932), who proved that with one exceptional case the structure of G can be recovered completely from its Line graph. The Line graph is defined as follows. The Line graph [72, 81] of G denoted by L(G) is the intersection graph of the edges of G, representing each edge by the set of its two end vertices. Otherwise L(G) is a graph such that Each vertex of L(G) represents an edge of G. Two vertices of L(G) are adjacent if their corresponding edges share a common end point in G. Example Figure 1(a): Graph G Figure 1(b): Line graph L(G) 62

2 5.2 Properties of Line Graph [46,76] Properties of a graph G that depend only on adjacency between edges may be translated into equivalent properties in L(G) that depend on adjacency between vertices. For instance, a matching in G is a set of edges no two of which are adjacent, and corresponds to a set of vertices in L(G) no two of which are adjacent, i.e. an independent set. Some of the important properties of Line graph are as follows: The Line graph of a connected graph is connected. If G is connected, it contains a path connecting any two of its edges, which translates into a path in L(G) containing any two of the vertices of L(G). However, a graph G that has some isolated vertices, and is therefore disconnected, may nevertheless have a connected Line graph. A maximum independent set in a Line graph corresponds to maximum matching in the original graph. Since maximum matching may be found in polynomial time. so may the maximum independent sets of Line graphs, despite the hardness of the maximum independent set problem for more general families of graphs. The edge chromatic number of a graph G is equal to the vertex chromatic number of its Line graph L(G). The Line graph of an edge-transitive graph is vertex-transitive. If a graph G has an Euler cycle, that is, if G is connected and has an even number of edges at each vertex, then the Line graph of G is Hamiltonian. Line graphs are claw-free graphs, graphs without an induced subgraph in the form of a three-leaf tree. The Line graphs of trees are exactly the claw-free block graph. T = T(G) is Eulerian if and only if the Line graph L(G) is Eulerian. 5.3 b-chromatic Number of Line Graph of Star Graph K 1,n Theorem For every n,ϕ[l(k 1,n )] = n 63

3 Consider the Line graph K 1,n. The Line graph of K 1,n is a Complete graph with n-vertices. We know that the b-chromatic number of Complete graph K n requires n-colours for producing a b-colouring. Therefore ϕ[l(k 1,n )] = n. Example Figure 2(a): K 1,5 Figure 2(b):ϕ[L(K 1,5 )] =5 5.4 b-chromatic Number of Line Graph of Cycle Theorem For any Cycle C n, φ[l(c n )] =3 L(C n ) C n.. By Theorem [3.3.1], we have φ[l(c n )] =3 Example Figure 3: φ[l(c 8 )] =3 64

4 5.5 b-chromatic Number of Line Graph of Pan Graph Theorem The b-chromatic number of every Line graph of Pan graph is tricolourable. By definition, the n-pan graph is the graph obtained by joining the Cycle graph C n to K 1 with a bridge. Consider the Line graph of Pan graph. By the definition of Line graph, the vertex set of Line graph of Pan graph corresponds to edge set of Pan graph. Consider the Line graph of Pan graph, we see that every Line graph of Pan graph is a union of cycle C n with K 3. First we assign the colour to complete graph K 3, by colouring procedure it requires three colours for producing a b-chromatic colouring. If we assign any new colour to the cycle C n, then it does not produce b-chromatic colouring because the complete graph K 3 do not realizes the new colour. Thus, by the colouring procedure the b-chromatic number of every Line graph of Pan graph is three. Hence by very construction the above said colouring is maximal. Example Figure 4: Pan graph with four vertices Observation The Line graph of Pan Graph is a Diamond graph (when n=4 ). 65

5 5.6 b-chromatic Number of Line Graph of Complete Bipartite Graph Theorem For the Line graph K m,n, ϕ[l(k m,n )] = Max{m,n} for every m,n 2 Let K m,n be the Complete Bi-partite graph with bipartition (X,Y) where X={v 1,v 2,v 3..v m } and Y={u 1,u 2,u 3..u n }. Consider the Line graph of K m,n i.e. L(K m,n ). Let v ij be the edge between the vertex v i and u j for i=1,2,3..m, j =1,2,3..n i.e. v i u j = {v ij: 1 i m, 1 j n}. By the definition of the Line graph, edges in K m,n corresponds to the vertices in L(K m,n ) i.e. V [L(K m,n )]= {v ij : 1 i m, 1 j n}. Note that for each i, we say that < v ij: j=1,2,3..n > is a complete graph of order n. Also we say for each j, < v ij : i=1,2,3..m > forms a complete graph of order m. Clearly the number of cliques in L(K m,n ) is m+n. Case 1 when m < n By observation in L(K m,n ), we have K n > K m. Consider the colour class C={c 1,c 2,c 3..c n }. Now assign a proper colouring to the vertices as follows. First assign the colours to the vertices v ij (1 i m, :1 j n) as follows. Here < v mj : j=1,2,3..n > for m=1 forms a complete graph of order n. Assign c j to v 1j for j=1,2,3..n, which produces a b-chromatic colouring. Suppose if we assign any new colour to the remaining complete graph < v ij : i=2,3..n, j=1,2,3..n >, it contradicts the definition of b-chromatic colouring because the remaining complete graphs does not realize the new colour. Hence to make the colouring as b-chromatic one, assign the colouring as follows. First assign the colour c i to the vertex v ij when j=1, i=1,2,3..m and assign c j to v ij s when i=1, j=1,2,3..n. Next for i=2..3..m and j=2,3 n, assign the colour c i+j-1 to v ij s when i+j n+1 and assign c i+j-(n+1) when i+j > n+1. Now all the n vertices realize its own colour, which produces a b-chromatic colouring. Thus by the colouring procedure the above said colouring is maximum and b-chromatic. Therefore φ[l(k m,n )] =n. 66

6 Example Figure 5:ϕ[L(K 3,4 )] = 4 Case 2 when m > n In L(K m,n ) we have K m > K n. Consider the colour class C={c 1,c 2,c 3..c n }. Now assign a proper colouring to the vertices v ij (1 i m, 1 j n) as follows. Here <v ni: i=1,2,3..m> for n=1 forms a complete graph of order n. Assign c i to v i1 for i=1,2,3..m, which produces a b-chromatic colouring. Suppose if we assign any new colour to the remaining complete graph <v ij : i=2,3,..m, j=1,2,3..n>, it contradicts the definition of b-chromatic colouring. Hence to make the colouring as b-chromatic one, assign the colouring to the vertices as follows. First assign the colour c i to v ij when j=1, i=1,2 m and assign colour c j to v ij s when i=1, j=1,2,3..n. Next for i=2,3..m and j= 2,3 n, assign c i+j-1 to v ij when i+j m+1 and assign c i+j-(m+1) when i+j > m+1. Now here all m vertices realize its own colour, which produces a b-chromatic colouring. Thus by the colouring procedure the above said colouring is maximum and b-chromatic. Therefore φ[l(k m,n )] =m. 67

7 Example Figure 6:ϕ[L(K 4,3 )] = 4 Case 3 when m = n In this case, K m will become K n. Clearly the number of cliques in L(K n,n ) is 2n. By following the procedure given in above cases, we have φ[l(k n,n )] =n. Example Figure 7:ϕ[L(K 4,4 )] = 4 From all the above cases, φ[l(k m,n )] =Max{m,n}. 68

8 5.6.2 Structural Properties of Line Graph of Complete Bipartite Graph Number of vertices in L(K m,n ) = m+n Number of edges in L(K m,n )= (n+m-2) Maximum degree of L(K m,n ) is = m+n-2 Minimum degree of L(K m,n ) is δ = m+n Corollary Every Line graph of K m,n is a m+n-2 regular graph Theorem For any Complete Bipartite graph K m,n, the number of edges in L[K m,n ] = (n+m-2) q L[K m,n ] = Number of edges in all K n + Number of edges in all K m = m q(k n )+ n q(k m ) = m + n = m ( ) = (n-1+m-1) = (n+m-2) +n Therefore q[l(k m,n )] = (n+m-2) ( ) 5.7 b-chromatic Number of Line Graph of Double Star Graph Theorem ϕ[l(k 1,n,n )] =n for every n 2. Consider the Double Star graph K 1,n,n. It is the tree obtained from the Star graph K 1,n by adding a new pendant vertex to the existing n pendant vertices. Here K 1,n,n is a Double star graph with v as the root vertex along with the vertex set v 1, v 2,..., v n and v 1, v 2,..., v n together with 69

9 the edges u 1,u 2, u n and u 1,u 2, u n. Now construct the Line graph of K 1,n,n. By the definition of Line graph, the edge set in K 1,n,n corresponds to the vertex set of L(K 1,n,n ) respectively. i.e. V [L(K 1,n,n )] = {u i / 1 i n} {u i / 1 i n}. In L(K 1,n,n ) we say that the vertices u 1,u 2, u n induces a clique of order n(say K n ). Also we say that the vertices u i is adjacent with the vertex u i for i = 1,2,3.n. we assign a proper colouring to these vertices as follows. Consider a colour class C= {c 1,c 2, c n }. Assign the colour c i to the vertex u i for i = 1,2,3.n, here all the vertices in u i for i=1,2,3..n realizes its own colour. Hence the colouring is b-chromatic colouring. Next we assign the colour c n+1 to all u i for i=1,2,3.n, due to the above mentioned adjacency condition the vertex set {u i : i=1,2,3..n } does not realizes the colour c n+1. So there is a possibility of assigning only the preused colours to the vertices u 1,u 2, u n. Note that rearrangement of colours also does not accommodate the new colour class. Thus by the colouring procedure the above said colouring is maximum and b-chromatic colouring. Example Figure 8: K 1,n,n 70

10 Figure 9:ϕ[L(K 1,5,5 )]= Structural Properties of Line graph of Double Star Graph Number of vertices in L(K 1,n,n ) = 2n Number of edges in L(K 1,n,n ) = () Maximum degree of L(K 1,n,n ) is = n Minimum degree of L(K 1,n,n ) is δ = 1 n vertices are with degree n and Theorem q[l(k 1,n,n )]= q[l(k 1,n,n )] = Number of edges in K n + Number of edges not in K n = q(k n ) + Number of edges not in K n = + n = (1) +n 2 = ( ) 71

11 = = () = Therefore q[l(k 1,n,n )] = Theorem For every integer n, ϕ [L(K 1,n,n )] =ϕ[l(k 1,n )] = n The proof follows from the Theorem and b-chromatic Number of Line Graph of Fire Cracker Graph Theorem ϕ[l(f 2,k )] = k for k 2 Let G = F 2,k be the Fire cracker graph. By definition, (2,k) Fire Cracker graph is obtained by concatenation of 2,k stars by linking one leaf from each. Consider the Line graph of F 2,k. Let S be the vertex adjacent with both v and v. Here the vertex v along with v 1,v 2,..,v k-1, induces a clique of order k also the vertex v with v 1, v 2..., v k-2 induces another clique of order k. Thus in L(F 2,k ), we find two copies of mutually disjoint Complete subgraphs. Consider a colour class C= {c 1,c 2, c k }. Now assign a proper colouring to these vertices as follows. First assign the colour c 1 to the vertex v and c i+1 to the vertices v 1, v 2,..., v k-2 for i=1,2,3...k which produces a b-chromatic colouring. Next suppose if we assign any new colours to v and v i for i=1,2 k-1 then it will not produce a b-chromatic colouring. Similarly if we assign any colour to the root vertex S, again it fails to produce the b-chromatic colouring. Because here the vertex set v and v are mutually disjoint to each other. Thus the only possibility is to assign the same colour which we already assigned for the vertices v and v i for i=1,2,3..k-1 such as c 1 to v and c i+1 to v i for i=1,2,3 k-1 and the colour c 2 72

12 to the root vertex s. Now all the vertices vv i and vv i realizes its own colour, which produces a b-chromatic colouring. Thus by the colouring procedure the above said colouring produces a maximum and b-chromatic colouring. Example Figure 10: F 2,5 Figure 11: φ[l(f 2,5 )] = Structural Properties of Line Graph of Fire Cracker Graph Number of vertices in L(F 2,k ) = 2k+1 Number of edges in L(F 2,k )= 2( K+1 C 2 )+ 2 Maximum degree of L(F 2,k ) is = k Minimum degree of L(F 2,k ) is δ = 2 73

13 5.8.3 Theorem q[l(f n,k )] = n( K+1 C n )+ n where n=2 q[l(f n,k )] = Number of edges in all K k+1 + Number of edges not in any of the K k+1 = n q(k k+1 ) + Number of edges not in any of the K k = n () +n = n( k+1 C n )+n Therefore q[l(f n,k )] = n( K+1 C n )+ n Results under Observation ϕ[l(f 3,k )] = k for all k 2. ϕ[l(f 4,k )] = k for all k 2. ϕ[l(f 5,k )] = k for all k 2 and so on. 5.9 b-chromatic Number of L[B n,n ] and L[P m ӨS n ] Theorem For every n 2, φ[l(b n,n )] = n+1 Consider the Bistar B n,n. By definition of Bistar, let u 1,u 2,.u n be the n pendant edges attached to the vertex u and v 1,v 2,.v n be the another n pendant edges attached to the vertex v. For i=1,2,3..n, let u i be the edge between the vertex uu i and v i is the edge between the vertex vv i and w be the edge between u and v i.e. uu i = u i,vv i = v i and uv= w. Here w is adjacent with both the vertices u i and v i for i=1,2,3 n. Consider the Line graph of B n,n. By the definition of Line graph, the edge set of Bistar corresponds to the vertex set of L(B n,n ). In L[B n,n ] the vertices u i (i=1,2 n) along with w forms a complete graph of order n+1. Also we see that the vertices v i (i=1,2 n) together with w forms another complete graph of order n+1. 74

14 Thus it contains two copies of edge disjoint complete graph of order n+1 i.e. let K i n+1 be the cliques in L[B n,n ] for i 2. Number these complete sub graphs as K 1 2 n+1 and K n+1. Consider the colour class C={c 1,c 2,c 3 c n,c n+1 }. First assign the colour c i to u i for 1 i=1,2,3..n and c n+1 to w. Here in K n+1, vertices u i (i=1,2,3..n) and the vertex w realizes its own colour which produces a b-chromatic colouring. Next assign the colour c n+i+1 to the vertices v i (i=1,2,3..n) of K 2 n+1, here other than the vertex w none of the vertices u i and v i realizes the new colours, which does not produce a b-chromatic colouring because the vertices u i and v i are mutually disjoint. So we cannot assign any new colour to the vertices v i (i=1,2,3..n). Thus to make the colouring as b-chromatic one, we should assign only the same set of colours to v i (i=1,2,3..n) which we already assigned for u i (i=1,2,3 n). Now all the vertices u i,v i and w realizes its own colour, which produces a b-chromatic colouring. Thus by the colouring procedure the above said colouring is maximum and b-chromatic. Example Figure 12: φ [L(B 5,5 )] =6 75

15 5.9.2 Structural Properties of Line Graph of Bistar Number of vertices in L(B n,n ) = 2n+1 Number of edges in L(B n,n ) = n(n+1) Maximum degree of L(B n,n ) = 2n Minimum degree of L(B n,n ) = n-1 In L(B n,n ) there are two copies of edge disjoint K n+1. 2n vertices of degree n and 1 vertex of degree 2n Theorem For every n,m 2, φ[ L(P m ӨS n )] = n+1 Consider the tree P m ӨS n. Let its vertex set be defined as V={v 1,v 2,v 3..v 2n+m } and the edge set be defined as E={e 1,e 2,e 3,e n,e n+1.e n+m-1,e n+m,e n+m+1, e 2n+m-1 }. Consider the Line graph of P m ӨS n. By the definition of the Line graph, edge set in P m ӨS n corresponds to the vertex set of L[P m ӨS n ] i.e. V [L(P m ӨS n )] = {e i / 1 i 2n+m-1}. Here in L(P m ӨS n ), the vertices e 1,e 2,e 3,e n along with e n+1 forms a complete graph of order n+1 namely K 1 and the vertices e n+m,e n+m+1, e 2n+m-1 along with e n+m-1 forms another complete graph of order n+1 namely K 2. Thus we have two copies of mutually disjoint subgraphs. Consider the colour class C={c 1,c 2,c 3, c 4..c n,c n+1 }. Now assign a proper colouring to the above vertices as follows. By Theorem and by the colouring procedure we can assign same set of n+1 colours to the both the graphs K 1 and K 2 and assign any existing colours to the remaining vertices. Thus by the colouring procedure the above said colouring is maximum and b-chromatic. 76

16 Example Figure 13: L(P 5 ӨS 5 ) = Structural Properties of L(P m ӨS n ) Number of vertices in L(P m ӨS n )= 2n+m-1 Number of edges in L(P m ӨS n )= n 2 +n+m-1. The Maximum degree of L(P m ӨS n ) is = n+1. The Minimum degree of L(P m ӨS n ) is δ =2. In L(P m ӨS n ), there are 2n vertices of degree n, 2 vertices of degree n+1, and remaining vertices of degree 2. 77

17 5.10 b-chromatic Number of Line Graph of Central Graph of Complete Graph Theorem For any Complete graph K n, ϕ{l[c(k n )] } = n for n 2 Let K n be the Complete graph on n vertices and edge set of K n contains exactly edges. Consider the Central graph of Complete graph K n. By the definition of Central graph, let v ij be the newly introduced vertex in the edge connecting vertex v i v j in C(K n ). Let v i v ij =e ij and v ij v i =e ji. Clearly {e ij: 1 i n-1, i+1 j n}. Here we considered only undirected graph so that we have e ij = e ji. Consider the Line graph of Central graph of the Complete graph K n. By the definition of Line graph the edge set in C(K n ) corresponds to the vertex set of L[C(K n )]. Under observation we obtain n-copies of vertex disjoint K n-1 complete subgraphs. Now number these complete subgraphs in anticlockwise direction namely K 1 n-1, K 2 n-1.k n n-1 for i= 1,2,3, n i.e. K i n-1 be the cliques in L[C(K n )]. Therefore we say ϕ{l[c(k n )]} n-1. Assign a proper colouring to the above vertices as follows. Consider the colour class C = {c 1,c 2, c n, }. Assign the colour c 1,c 2,c 3..c n-1 to K i n-1 for i=1 and assign the colour c n to the vertices in the remaining subgraph K i n-1(i=2,3..n) in which the vertex is adjacent with K 1 n-1 and all the other vertices in K i n-1 for i=2,3..n to be coloured with the existing colours other than the colour c n. Now all the vertices realize its own colour which produces a b-chromatic colouring. Thus by the colouring procedure the above said colouring is maximum and b-chromatic. 78

18 Example Figure 14:ϕ{L[C(K 6 )] } = Structural Properties of Line Graph of Central Graph of Complete Graph Number of vertices in L[C(K n )] = n(n-1) Number of edges in L[C(K n )]= ( ) Maximum degree of L[C(K n )] is = n-1 Minimum degree of L[C(K n )] is δ = n-1 L[C(K n )] is Hamiltonian and Eulerian 79

19 Theorem ϕ{l[c(k 3 )]} =ϕ[c(k 3 )] By observation, we find that C(K 3 ) is C 6 and its Line graph is also C 6, since L(C n ) C n. Therefore by Theorem [3.3.1], we have ϕ{l[c(k 3 )]} =ϕ[c(k 3 )] 5.11 b-chromatic Number of Central Graph of Line Graph of Cycle and Line Graph of Central Graph of Cycle Theorem For any Cycle C n of length n 5, n=5x+r,0 < 5 ( +1) when 0 ϕ{c [L(C n )]}= when = 0 Here L(C n ) C n. By the property of Central graphs, C[L(C n )] C[C n. ]. Therefore by Theorem 4.3.1, we have ϕ{c [L(C n )]} = ϕ$%(% )& ( +1) when 0 = when = Structural Properties of Line Graph of Central Graph of Cycle Number of vertices in L[C(C n )] = () Number of edges in L[C(C n )] = n' +1( Maximum degree of L[C(C n )] is = 2(n-2) Minimum degree of L[C(C n )] is δ = n-1 2n vertices has degree n-1 and ( )) L[C(C n )] contains n copies of edge disjoint K n-1. vertices has degree 2(n-2) Theorem For any n 2, φ{l[c(c n )]} =n 80

20 5.12 b-chromatic Number of Line Graph of Central Graph of K 1,n Theorem For any Complete Bipartite graph K 1,n, ϕ{l[c(k 1,n )]} = n+1 for every n 2 Consider the Star graph K 1,n with vertices v 1, v 2,...,v n, v where v 1, v 2,..., v n be the pendant vertices of K 1,n and let v be the root vertex of K 1,n adjacent to v i for 1 i n. By the definition of the Central graph, each edge vv i for 1 i n of K 1,n is subdivided by a newly introduced vertex v i in C(K 1,n ) i.e. v 1, v 2,..., v n are the vertices of subdivision at each of the edges vv 1, vv 2, vv n of C(K 1,n ). By definition of C(K 1,n ), the vertices v 1, v 2,..., v n, induces a clique of order n (say K n ) in every C(K 1,n ). Let u i (i=1,2..n) be the edge between the vertex vv i and u i be the edge between the vertex v i v i. Now consider the Line graph of Central graph of K 1,n i.e. L[C(K 1,n )]. Here the edge set in C(K 1,n ) corresponds to the vertex set of L[C(K 1,n )]. Here in L[C(K 1,n )],the vertices <u i: 1 i n>forms a complete graph of order n. Also we see that each edge u i is adjacent with u i for i=1,2,3..n. In L[C(K 1,n )] we find the remaining n vertices are of degree n and nc 2 vertices of degree 2n-2. Now assign a proper colouring to the above vertices as follows. Consider the colour class c 1,c 2, c n,c n+1. First assign the colour c i to u i for i=1,2,3..n and assign the colour c n+i to u i for i=1,2,3..n. Here the vertices u i do not realize the colour c n+i, which does not produce b-chromatic colouring. To make the colouring as b-chromatic one, assign the colour to the remaining vertices as follows. Assign the colour c n+1 to u i for i=1,2,3.n. Here each u i is adjacent with with n vertices. Consider any arbitrary vertex u i for i=1,2,3..n. Assign the colour c 1,c 2, c n to the vertices adjacent with u i other than the colour assigned to u i. Note that rearrangement of colours also does not accommodate new colour class. Thus by colouring procedure the above said colouring is maximum and b-chromatic colouring. 81

21 Example Figure 15: K 1,n Figure 16:C(K 1,n ) 82

22 Figure 17:ϕ[L{C(K 1,3 )}] = Structural Properties of Line Graph of Central Graph of Star Graph Number of vertices in L[C(K 1, n )] = ()) Number of edges in L[C(K 1, n )]= * Maximum degree of L[C(K 1, n )] is = 2(n-1) Minimum degree of L[C(K 1, n )] is δ = n Remark L[C(K 1,1 )] = C 3 L[C(K 1,2 )] = C 5 83