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The Rainbow Connection of Windmill and Corona Graph

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The Rainbow Connection of Windmill and Corona Graph Applied Mathematical Sciences, Vol. 8, 2014, no. 128, 6367 - 6372 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.48632 The Rainbow Connection of Windmill and Corona Graph Yixiao Liu Department of Mathematics, Dalian Maritime University Dalian, P.R. China, 116026 Zhiping Wang∗ Department of Mathematics, Dalian Maritime University Dalian, P.R. China, 116026 ∗Corresponding author Copyright c 2014 Yixiao Liu and Zhiping Wang. This is an open access article dis- tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The rainbow connection number of G, denoted by rc(G), is the small- est number of colors needed to color its edges, so that every pair of its vertices is connected by at least one path in which no two edges are colored the same. A rainbow u − v geodesic in G is a rainbow path of length d(u; v), where d(u; v) is the distance between u and v. The graph G is strongly rainbow connected if there exists a rainbow u − v geodesic for any two vertices u and v in G. The strong rainbow connection num- ber src(G) of G is the minimum number of colors needed to make G strongly rainbow connected. (n) In this paper we determine the exact values of the windmill graph Km . Moreover, we compute the rc(G ◦ H) where G or H is complete graph Km or path P2 with m is an integer. These graphs we studied have progressive connection on structure. Mathematics Subject Classification: 05C15, 05C40 Keywords: (strong) rainbow connection number, windmill graph, corona graph 6368 Yixiao Liu and Zhiping Wang 1 Introduction An edge coloring of a graph is a function from its edges set to the set of natural numbers. The concept of rainbow connectivity was recently introduced by Chartrand et al. in [1]. A path is rainbow if no two edges of it are colored the same. An edge-coloring graph G is rainbow connected if any two vertices are connected by a rainbow path. The minimum k for which there exists a rainbow k-coloring of the edges of G is the rainbow connection number rc(G) of G. For two vertices u and v of G, a rainbow u−v geodesic in G is a rainbow u−v path of length d(u; v), where d(u; v) is the distance between u and v. The graph G is strongly rainbow-connected if G contains a rainbow u − v geodesic for every two vertices u and v of G. The minimum k for which there exists a coloring c of the edges of G such that G is strongly rainbow-connected is the strong rainbow connection number src(G) of G. Thus, rc(G)≤src(G) for every connected graph G. Furthermore, if G is a nontrivial connected graph of size m whose diameter denoted by diam(G), then diam(G) ≤ rc(G) ≤ src(G) ≤ m. The rainbow connection of fan, sun and some corona graphs have been studied in [2, 3]. There have been some results on the (strong) rainbow connection numbers of graphs (see [1, 4]). In this paper, we will give the (strong) rainbow connection numbers of the windmill graph and corona graph. 2 Preliminary Notes At the beginning of this paper, we first introduce some preliminary notes used in the paper. (n) Definition 2.1. [5] The windmill graph Km is the graph consisting of n copies of the complete graph Km with a vertex in common. Definition 2.2. [3] The corona G ◦ H of two graphs G and H (where jV (G)j = m and jV (H)j = n) is defined as the graph G obtained by taking one copy of G and copies of H, called H1;H2;:::;Hm, and then joining by a line the i'th vertex of G to every vertex in the i'th copy of H. 3 Results and Discussion At the beginning of this part, we will find the rainbow connection number and the strong rainbow connection number of windmill graph. Theorem 3.1. For positive integers m and n we have that 8 > 1; n = 1; (n) < rc(Km ) = 2; n = 2; :> 3; n ≥ 3: The rainbow connection of windmill and corona graph 6369 (n) the strong rainbow connection number of the windmill graph is src(Km ) = n. Proof. Let v0 be the center vertex and uij be the vertices of the i'th complete graph(i 2 f1; : : : ; ng j 2 f1; : : : ; m − 1g). First, we show the proof of the rainbow connection number. Since the proofs of n = 1; 2 are obvious, we consider the situation where n ≥ 3. Because of (n) (n) (n) (n) diam(Km ) = 2, rc(Km ) ≥ diam(Km ) = 2. Assume rc(Km ) = 2, then (n) there exists a rainbow 2-coloring c : E(Km ) ! f1; 2g. Without loss of generality, we let c(v0u11) = 1. u11 − v0 − u21 is the only u11 − u21 path (n) of length 2 in Km . There is a rainbow path from u11 to u21 if and only if c(v0u21) = 2. However, there is no rainbow path from u21 to u31 in this case. (n) (n) Hence, rc(Km ) ≥ 3. We distinguish two cases to show that rc(Km ) = 3. Before giving proof, we should explain a symbol. First, each complete graph i is labeled according to clockwise. Then, Km is the symbol of the i'th complete graph. (3) Case 1 n = 3, to show that rc(Km ) ≤ 3, we provide a rainbow 3-coloring (n) i c : E(Km ) ! f0; 1; 2g defined by c(Km) = i − 1 (i 2 f1; 2; 3g). This implies that all the edges of the i'th complete graph are colored with the color i − 1. (3) Therefore, rc(Km ) = 3. (n) Case 2 n > 3, we define a 3-coloring c : E(Km ) ! f0; 1; 2g, let any three of the complete graphs be colored 0, 1, 2 respectively. Without loss of generality, k let c(Km) = k − 1 (k 2 f1; 2; 3g). Now, we color the rest of the complete graphs as follows: 8 c(v u ) = j mod 3; <> 0 ij c(uijuij+1) = f0; 1; 2g − c(v0uij) − c(v0uij+1); > : c(e) 2 f0; 1; 2g; for e 2 E(G) − fv0uijg − fuijuij+1g: Where i 2 f4; 5; : : : ; ng, j 2 f1; 2; : : : ; m − 1g. In this coloring, all the edges of a complete graph are colored with at least one color. So, each complete graph is rainbow connected under the coloring we have defined above. Meanwhile, each path from one vertex ui1j1 of a complete graph to a vertex ui2j2 of another complete graph is colored according to the following three conditions. See Fig. 1. (n) (n) This implies that rc(Km ) ≤ 3. Thus rc(Km ) = 3 (n ≥ 3). (n) Second, we show that src(Km ) = n. Since u11 − v0 − u21 is the only u11 − u21 (n) path with length d(u11u21) between u11 and u21 in Km , what if u11 − v0 − u21 is a geodesic, the edges v0 − u11, v0 − u21 need to be colored different. In the procedure, the edges v0 − ui1 (i 2 f1; : : : ; ng) have distinct colors. (n) src(Km ) ≥ n was established. Now we provide a strong rainbow n-coloring ∗ (n) (n) ∗ (i) c : E(Km ) ! f1; 2; ··· ; ng of Km defined by c (Km ) = i (i 2 f1; : : : ; ng). (n) This suggests src(Km ) ≤ n. Hence the strong rainbow connection number is (n) src(Km ) = n. 6370 Yixiao Liu and Zhiping Wang Figure 1: The three cases of the rainbow path When the sharing vertex of the windmill graph replaced by the complete graph, we can derive some special corona graphs. In the following, we deter- mine the rainbow connection numbers of some corona graphs for combination of some complete graphs and P2. Theorem 3.2. The rainbow connection number of G ◦ H is 8 3; m = 3; > for (G ∼= K and H ∼= P ); <> 4; m ≥ 4: m 2 rc(G ◦ H) = 3; m = 3; > for (G ∼= K and H ∼= K with n ≥ 3): :> 4; m ≥ 4: m n Proof. We consider two cases. ∼ ∼ Case 1 For G = Km and H = P2. Since diam(Km◦P2) = 3, then rc(Km◦P2) ≥ diam(Km◦P2) = 3. For m = 3, we define a rainbow 3-coloring c : E(K3◦P2) ! f1; 2; 3g of Km ◦ P2 as follows: ( c(vmum1) = c(vmum2) = c(um1um2) = m; m 2 f1; 2; 3g; c(vivj) = f1; 2; 3g − fi; jg; i 6= j and i; j 2 f1; 2; 3g: This implies that rc(K3 ◦P2) ≤ 3. Thus rc(K3 ◦P2) = 3. For m ≥ 4, if rc(Km ◦ P2) = 3, there will always exists a non-rainbow path uisvi − vi − vj − vjujt that the edges uisvi and vjujt are colored with the same color. Thus rc(Km◦P2) > 3. We can construct a rainbow 4-coloring c : E(Km ◦ P2) ! f1; 2; 3; 4g defined by: 8 c(v u ) = c(v u ) = c(u u ) = m; for m 2 f1; 2; 3; 4g; > m m1 m m2 m1 m2 > > c(vmum1) = 1; > > c(v u ) = 3; for m 2 f5; 6; : : : ; ng; <> m m2 c(um1um2) = 2: > > c(vivj) 2 f1; 2; 3; 4g − fi; jg; for i 6= j i; j 2 f1; 2; 3; 4g; > > > c(vpvq) = 4; for 4 6= p 2 f1; : : : ; ng and q 2 f5; 6; : : : ; ng; > : c(v4vq) 2 f1; 2; 3g; for q 2 f5; 6; : : : ; ng: The rainbow connection of windmill and corona graph 6371 Figure 2: The case of all edges of E(Km ◦ P2) m ≥ 4 are different colored Similar to the theorem 3.1.
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