GRAPH OF ANY GRAPH WITH PATH

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1 CHAPTER 4 b-coloring OF CORONA GRAPHS In this Chapter, the author finds that the b-chromatic number on corona graph of any graph G of order n0. with path P n, cycle C n and complete graph K n. Finally, they generalized the b-chromatic number on corona graph of any two graphs, each one on n vertices. 4.1 b-chromatic NUMBER ON CORONA GRAPH OF ANY GRAPH WITH PATH Theorem Let G be a simple graph on n vertices. Then n + 1; for n 3, ϕ(g P n ) = n; for n > 3. Proof. Let V (G) = {v 1, v 2,...,v n } and V (P n ) = {u 1, u 2,...,u n }. Let V (G P n ) = {v i : 1 : i n} {u ij : 1 i n; 1 j n}. By the definition of corona graph, each vertex of G is adjacent to every vertex of a copy of P n. i.e., every vertex v i V (G) is adjacent to every vertex from the set {u ij : 1 j n}. Assume first that n > 3 and assign the following n-coloring for G P n as b-chromatic: For 1 i n, assign the color c i to v i. 65

2 66 For 1 i n, assign the color c i to u 1i, i 1. For 1 i n, assign the color c i to u 2i, i 2. For 1 i n, assign the color c i to u 3i, i 3. For 1 i n, assign the color c i to u 4i, i For 1 i n, assign the color c i to u ni, i n. For 1 i n, assign to vertex u ii one of the allowed colors - such color exists, because 2 deg(u ii ) 3 and n > 3. Let us assume that ϕ(g P n ) is greater than n, i.e. ϕ(g P n ) = n + 1, n > 3, there must be at least n + 1 vertices of degree n in G P n, all with distinct colors, and each adjacent to vertices of all of the other colors. But then these must be the vertices v 1, v 2,...v n, since these are only ones with degree at least n. This is the contradiction, b-coloring with n+1 colors is impossible. Thus, we have ϕ(g P n ) n. Hence, ϕ(g P n ) = n, n > 3. Note that ϕ(g P 1 ) = 2 for graph G on one vertex and ϕ(g P 2 ) = 3 for graph G on two vertices. Indeed, let us notice that such graph G P 2 inculdes K 3. Now, let us define b-coloring of G P 3 ( V (G) ) = 3 with four colors in the following way: for 1 i 3, assign the color c i to v i, for 1 l 3, assign the color c l to u 1l, l 1, for 1 l 3, assign the color c l to u 2l, l 2, for 1 l 3, assign the color c l to u 3l, l 3 and for 1 l 3, assign the color c 4 to u ll. Therefore, ϕ(g P 3 ) 4. Let us assume that ϕ(g P 3 ) is greater than 4, i.e. ϕ(g P 3 ) = 5, there must be at least 5 vertices of degree 4 in G P 3, all with distinct colors, and each adjacent to vertices of all of the other colors. But then these must be the vertices v 1, v 2 and v 3, since these are only ones with degree at least 4. This is the

3 67 contradiction, b-coloring with 5 colors is impossible. Thus, we have ϕ(g P 3 ) 4. Therefore,ϕ(G P 3 ) = 4. Hence, ϕ(g P n ) = n + 1, n 3. Figure 4.1: b-coloring of G P 3 with four colors for G = P b-chromatic NUMBER ON CORONA GRAPH OF ANY GRAPH WITH CYCLE Theorem Let G be a simple graph on n vertices, n > 3. Then ϕ(g C n ) = n. Proof. Consider the coloring of G P n n > 3 introduced on the proof of Theorem An easy check shows that this coloring is a b - coloring of G C n n > 3. Thus, we have ϕ(g C n ) n. Hence, ϕ(g C n ) = n, n > 3. Note that ϕ(g C 3 ) = 4, since graph G C 3 includes graph K 4.

4 68 Figure 4.2: b-coloring of G C 3 with four colors for G = P b-chromatic NUMBER ON CORONA GRAPH OF ANY GRAPH WITH COMPLETE GRAPH Theorem Let G be a simple graph on n vertices. Then ϕ(g K n ) = n + 1. Proof. Let V (G) = {v 1, v 2,...,v n } and V (K n ) = {u 1, u 2,...,u n }. Let V (G K n ) = {v i : 1 i n} {u ij : 1 i n; 1 j n}. By the definition of corona graph, each vertex of G is adjacent to every vertex of a copy of K n. i.e.,every vertex v i V (G) is adjacent to every vertex from the set {u ij : 1 j n}. Assign the following n + 1-coloring for G K n as b-chromatic: For 1 i n, assign the color c i to v i. For 1 l n, assign color c l to u 1l, l 1. For 1 l n, assign color c l to u 2l, l 2.

5 69 For 1 l n, assign color c l to u 3l, l 3. For 1 l n, assign color c l to u 4l, l For 1 l n, assign the color c l to u nl, l n. For 1 l n, assign the color c n+1 to u ll. Therefore, ϕ(g K n ) n + 1. Let us assume that ϕ(g K n ) is greater than n+1, i.e., ϕ(g K n ) = n+2, there must be at least n + 2 vertices of degree n + 1 in G K n, all with distinct colors, and each adjacent to vertices of all of the other colors. But then these must be the vertices v 1, v 2,...v n, since these are only ones with degree at least n + 1. This is the contradiction, b-coloring with n + 2 colors is impossible. Thus, we have ϕ(g K n ) n + 1. Hence, ϕ(g K n ) = n b-chromatic NUMBER ON CORONA GRAPH OF STAR GRAPH WITH PATH Theorem Let n be a positive integer. Then ϕ(k 1,n P n ) = n + 1. Proof. Let V (K 1,n ) = {v 1, v 2,...,v n+1 } and V (P n ) = {u 1, u 2,...,u n }. Let v 1 be the central vertex of the star, v 1 is adjacent to each {v i : 2 i n}. Let V (K 1,n P n ) = {v i : 1 i n + 1} {u ij : 1 i n + 1; 1 j n}. By the definition of corona graph, each vertex of K 1,n is adjacent to every vertex of the corresponding

6 70 copy of P n. i.e., every vertex v i V (G) is adjacent to every vertex from the set {u ij : 1 j n}. Assign the following n + 1-coloring for K 1,n P n as b-chromatic: For 1 i n + 1, assign the color c i to v i. For 1 l n, assign color c l to u 1l, l 1. For 1 l n, assign color c l to u 2l, l 2. For 1 l n, assign color c l to u 3l, l 3. For 1 l n, assign color c l to u 4l, l For 1 l n, assign color c l to u nl, l n. For 1 l n, assign color c l to u n+1 l. For 1 l n, assign color c n+1 to u ll. Therefore, ϕ(k 1,n P n ) n + 1. Let us assume that ϕ(k 1,n P n ) is greater than n+1, i.e., ϕ(k 1,n P n ) = n+2, there must be at least n + 2 vertices of degree n + 1 in K 1,n P n, all with distinct colors, and each adjacent to vertices of all of the other colors. But then these must be the vertices v 1, v 2,...,v n+1, since these are only ones with degree at least n + 1. This is the contradiction, b-coloring with n + 2 colors is impossible. Thus, we have ϕ(k 1,n P n ) n + 1. Hence, ϕ(k 1,n P n ) = n + 1.

7 GENERALIZATION OF b-chromatic NUMBER ON CORONA GRAPH OF ANY TWO GRAPHS OF THE SAME OR- DER Theorem Let G and H be simple graphs, each one on n vertices. Then n ;if (H) < n 1, ϕ(g H) = n + 1; if (H) = n 1. Proof. Let V (G) = {v 1, v 2,...,v n } and V (H) = {u 1, u 2,...,u n }. Let V (G H) = {v i : 1 : i n} {u ij : 1 i n; 1 j n}. By the definition of corona graph, each vertex of G is adjacent to every vertex of a copy of H. i.e., every vertex v i V (G) is adjacent to every vertex from the set {u ij : 1 j n}. Let us rename vertices in ith copy of H in G H, i = 1, 2,...,n, in such a way that a vertex of maximum degree has a label u ii. Assign the following coloring for G H as b-chromatic: For 1 i n, assign the color c i to v i. For 1 i n, assign the color c i to u 1i, i 1. For 1 i n, assign the color c i to u 2i, i 2. For 1 i n, assign the color c i to u 3i, i 3. For 1 i n, assign the color c i to u 4i, i For 1 i n, assign the color c i to u ni, i n.

8 72 The following cases completes the proof: Case (i): (H) < n 1 For 1 i n, assign to vertex u ii one of the allowed colors - such color exists, because 1 deg(u ii ) < n. Therefore, ϕ(g H) n. Let us assume that ϕ(g H) is greater than n, i.e., ϕ(g H) = n + 1, there must be at least n + 1 vertices of degree n in G H, all with distinct colors, and each adjacent to vertices of all of the other colors. But then these must be the vertices v 1, v 2,...,v n, since these are only ones with degree at least n. This is the contradiction, b-coloring with n+1 colors is impossible. Thus, we have ϕ(g H) n. Hence, ϕ(g H) = n, if (H) < n 1. Case (ii): (H) = n 1 For 1 i n, assign the color c n+1 to u ii. Therefore, ϕ(g H) n + 1. Let us assume that ϕ(g H) is greater than n + 1, i.e., ϕ(g H) = n + 2, there must be at least n + 2 vertices of degree n + 1 in G H, all with distinct colors, and each adjacent to vertices of all of the other colors. But then these must be the vertices v 1, v 2,...,v n, since these are only ones with degree at least n. This is the contradiction, b-coloring with n + 2 colors is impossible. Thus, we have ϕ(g H) n + 1. Hence, ϕ(g H) = n + 1, if (H) = n 1.

9 73 Figure 4.3: b-coloring of G H = P6 W6 with seven colors. The case where (H) = n 1.