Rainbow Vertex-Connection Number of 3-Connected Graph

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1 Applied Mathematical Sciences, Vol. 11, 2017, no. 16, HIKARI Ltd, Rainbow Vertex-Connection Number of 3-Connected Graph Zhiping Wang, Xiaojing Xu and Yixiao Liu Department of Mathematics Dalian Maritime University Dalian, P.R. China, Copyright 2017 Zhiping Wang, Xiaojing Xu and Yixiao Liu. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A path in an edge colored graph is said to be a rainbow path if every edge in this colored with the same color. A vertex-colored graph G is rainbow vertex-connected if any pair of vertices in G are connected by a path whose internal vertices have distinct colors. The rainbow vertexconnection number of G denoted by rvc(g), is the smallest number of colors that are needed in order to make G rainbow vertex- connected. In this paper, we proved that rvc(g) 3(n+2) for 3-connected graph except to s + t 3. Mathematics Subject Classification: 0C1, 0C40 Keywords: subgraph rainbow, vertex-connection number, fan graph, connected 1 Introduction All graphs considered in this paper are undirected, finite and simple. Connectivity is perhaps the most fundamental graph-theoretic property. A natural and interesting quantifiable way to strengthen the connectivity requirement was recently introduced by Chartrand et al. [1]. Corresponding author

2 72 Zhiping Wang, Xiaojing Xu and Yixiao Liu Let G of order n be 3-connected graph. The rainbow vertex-connection number of G denoted by rvc(g), is the smallest number of colors that are need in order to make G rainbow vertex-connected (see [2]). The rainbow vertex-connection number have been studied in[3,4] and there have been some results on the rainbow vertex-connection number of graphs. Let H be the maximum connected subgraph of G. A rainbow u v in G is a vertex rainbow u v path. First we proved the existence of H. Then, we set up the coloring of the rainbow vertex-connection in many different cases. Through the discussion of that cases, finally, we proved that the rainbow vertex-connection number of 3-connected graph meet rvc(g) 3(n+2) except to s + t 3. 2 Preliminary Notes Before the formal research. First we give a fan-shaped theorem. The lemma will be used in the next process. Lemma 2.1 ([]) Let G be a k-connected graph. x a vertex of G, and let Y V x be a set at least k vertices of G. Then there exists a k -fan in G from x to Y,namely there exists a family of k internally disjoint (x, Y )-paths whose terminal vertices are distinct in Y. 3 Main results Theorem 3.1 If G of order n is 3-connected graph, in addition to a special case, then rvc(g) 3(n+2). Proof Let H be the maximum connected subgraph of G, thus rvc(h) 3h + 2, where h is the number of vertices in H. First, we prove the existence of H. If G contains a triangle C 3,then we can choose the triangle as H. For rvc(h) , thus the maximum connected subgraph H is existent. We first claim the existence of H. If G contains C k as a subgraph, then we take H = C k, since rvc(c 3 ) 3k < 3k Now we prove that h n 3. Let x 1, x 2, x 3, x 4 be the different vertex in outside of H. By lemma, every vertex has 3 paths with internal disjoint to H. (1) Assume every vertex of x 1, x 2, x 3, x 4 has 3 different adjacent vertex in H. Let v ij attached to x i be a vertex in H, where i = 1, 2, 3, 4, j = 1, 2, 3, 4. see Figure 1: We can add x 1, x 2, x 3, x 4 to H, and form a larger subgraph. The bigger subgraph H with h + 4 vertices. Now we use only two new colors 1 and 2 to color the 12 vertices.

3 Rainbow vertex-connection number of 3-connected graph 73 Figure 1: The first case 1) First coloring with color 1 coloring v 21, v 31, v 41 with color 2. There are vertex rainbow path x 1 x 2, x 1 x 3, x 1 x 4 in H. 2) Then coloring v 32, v 42 with color 2. There are vertex rainbow path x 2 x 3, x 2 x 4, x 3 x 4 in H. 3) Coloring the remaining 6 vertices with color 1 or color 2. Every pair of its vertices is connected by at least one rainbow vertex path in H. Thus rvc(h ) rvc(h) + 2 3h (h+4) + 2. That is in conflict with H which is the biggest connected subgraph. So the assumption is wrong. (2) There are at least one vertex with the following properties in x 1, x 2, x 3, x 4 (assume the vertex is x): The length of the path is at least 2, which is one of the three internally disjoint (x, H)-paths P 0, P 1, P 2. Furthermore, this four vertices satisfy the above property. We choose the vertex x such that one of the three paths has length one and the path is one of three paths with internal disjoint, i.e.p 0 = e 0. The length of path P 1, P 2 is as large as possible. Let P 1 = au 1 u 2 u s x, P 2 = xv 1 v 2 v t b where a, b H, u i, v j / H. We assume t s, and then t 1. We only consider 1 s + t 2. 1) s + t = 2 and P 1 = au 1 x, P 2 = xv 1 b Since there are at least 4 vertices outside of H, there exists at least one vertex different from x, u 1, v 1. Assume the vertex is x 1. By the choice of x 1, in addition to the following situation, if it don t use any vertex of H, then there is no (x 1, x)-path, (x 1, u 1 )-path and (x 1, v 1 )-path. The particular situation is one path of length two joining x to H through x 1 (assume P 3 = xx 1 c, c H). According to the fan lemma, there are 3 path P 0, P 1, P 2 of (x, H) with internal disjoint. By choosing x, the length of P 0, P 1, P 2 is only the following 4 possibilities: Case 1. P 0 = e 0, P 1 = e 1, P 2 = e 2 Now we just color P 0, P 1, P 2 of H with a new color 1, as is shown in Figure 2: +1 3(h+4) + 2. At this point, v(h ) = h+4. rvc(h ) rvc(h)+1 3h 1 Case 2. P 0 = e 0, P 1 = e 1, P 2 = x 1 v 1b We set up the coloring of the rainbow vertex-connection as shown in Figure 3: Now, v(h ) = h +. rvc(h ) rvc(h) + 2 3h (h+) + 2.

4 74 Zhiping Wang, Xiaojing Xu and Yixiao Liu Figure 2: The coloring in this case Figure 3: The coloring of the rainbow vertex-connection case 3. P 0 = e 0, P 1 = a u 1x 1, P 2 = x 1 v 1b We set up the coloring of rainbow vertex-connection as is shown in Figure 4: Figure 4: The coloring of the rainbow vertex-connection Now, v(h ) = h + 6. rvc(h ) rvc(h) + 2 3h (h+6) + 2. case 4. P 0 = e 0, P 1 = e 1, P 2 = x 1 v 1v 2b We set up the coloring of rainbow vertex-connection as is shown in Figure : Now, v(h ) = h + 6. rvc(h ) rvc(h) + 3 3h (h+6) ) s + t = 2 and P 1 = ax, P 2 = xv 1 v 2 b Since v 1 / H, there are 3 disjoint (v 1, H)-paths by the fan lemma.then there is at least one (v 1, H)-path P 3 in addition to the paths v 1 xa, v 1 v 2 b. By the choice of x, the length of P 3 must be at most 2. When l(p 3 ) = 2, it is different

5 Rainbow vertex-connection number of 3-connected graph 7 Figure : The coloring of the rainbow vertex-connection from case 1. When l(p 3 ) = 1, the path v 1 xa, v 1 v 2 b, P 3 are different from the structure of (1). It has been proven. 3) s + t = 1 Since t s, we have t = 1, s = 0. Assume P 1 = e, P 2 = xv 1 b. There are at least two distinct vertices in H different from x, v 1. Suppose this two vertices are x 1, x 2. Similarly, for i = 1, 2, there is no (x i, x)-path and (x i, v 1 )- path without using any vertex of H. So there are three internally disjoint (x 1, H)-paths P 0, P 1, P 2 and (x 2, H)-paths P 0, P 1, P 2. Case 1. If the length of all these paths is one, then we add x, v 1, x 1, v 2 to H, and form a larger subgraph H of order h + 4. We set up the coloring of the rainbow vertex-connection as is shown in Figure 6: Figure 6: The coloring of the rainbow vertex-connection Now, rvc(h ) rvc(h) + 1 3h (h+4) + 2. Case 2. Without loss of generality, we suppose one of the three (x 1, H)-paths P 0, P 1, P 2 has length 2. Let P 0 = e 0, P 1 = e 1, P 2 = x 1 v 1b. We add x, v 1, x 1, v 1 to H, and form a larger subgraph H of order h + 4. As is shown in Figure 7: Now,rvc(H ) rvc(h) + 1 3h (h+4) + 2. Now we have proved that h n 3 except the case s + t 3. If h = n 3, then rvc(g) rvc(h) + 2 3(n 3) = 3n+3 < 3(n+2). If h = n 2, then rvc(g) rvc(h) + 2 3n = 3n+6 < 3(n+2). If h = n 1, then rvc(g) rvc(h) + 1 3(n 1) = 3n+4 < 3(n+2). In conclusion, we can get the rainbow vertex-connection number of threeconnected graph for meet rvc(g) 3(n+2) except to s + t 3.

6 76 Zhiping Wang, Xiaojing Xu and Yixiao Liu Figure 7: The coloring of the rainbow vertex-connection 4 Conclusion In Theorem 3.1, we have calculated the rainbow vertex- connection number of 3-connected graph. If of order is 3-connected graph, in addition to s+t 3, then the graph G meet that rvc(g) 3(n+2). This paper discussed the rainbow vertex-connection number of 3-connected graph except a special case. It was not generalized. Finally, the rainbow vertex-connection number need further discussion. Acknowledgements. The work was supported by the Dalian Science and Technology Project Under contract No.201A11GX016. References [1] G. Chartrand, G. L. Johns, K. A. McKeon et al., Rainbow connection in graphs, Math. Bohem, 133 (2008), no. 1, [2] L. Chen, X. Li, H. Lian, Futher hardness results on the rainbow vertexconnection number of graphs, Theoretical Computer Science, 481 (2013), [3] X. L. Li, Y. T. Shi, On the rainbow vertex-connection, Discussiones Mathematicae Graph Theory, 33 (2013), no. 2, [4] X. L. Li, S. J. Liu, Rainbow vertex-connection number of 2-connected graphs. arxiv: vl[math.co], 2011.

7 Rainbow vertex-connection number of 3-connected graph 77 [] X. L. Li, Y. T. Shi, Rainbow connection in 3-connected graphs, Graphs and Combinatorics, 29 (2013), no., Received: January 11, 2017; Published: March 1, 2017