On total rainbow k-connected graphs

Abstract

A total-colored graph G is total rainbow connected if any two vertices are connected by a path whose edges and inner vertices have distinct colors. A graph G is total rainbow k-connected if there is a total-coloring of G with k colors such that G is total rainbow connected. The total rainbow connection number, denoted by trc(G), of a graph G is the smallest k to make G total rainbow k-connected. For n, k ≥ 1, define h(n, k) to be the minimum size of a total rainbow k-connected graph G of order n. In this paper, we prove a sharp upper bound for trc(G) in terms of the number of vertex-disjoint cycles of G. We also compute exact values and upper bounds for h(n, k).

Introduction

Any notation or terminology not defined here, follows that used in [2]. For a graph G, let V(G), E(G), n(G) and m(G), be the set of vertices, the set of edges, the order and the size of G, respectively.

Let G be a nontrivial connected graph on which an edge-coloring is defined, where adjacent edges may be colored the same. A path is rainbow if no two edges of it are colored the same. An edge-colored graph G is rainbow connected if any two vertices are connected by a rainbow path. A graph G is called rainbow k-connected if there is an edge-coloring of G with k colors such that G is rainbow connected. Chartrand et al. [5] defined the rainbow connection number of a connected graph G, denoted by rc(G), as the smallest k to make G rainbow k-connected. Clearly, rc(G) ≥ diam(G) where diam(G) denotes the diameter of G.

The rainbow connection number is not only a natural combinatorial measure, but also has applications to the secure transfer of classified information between agencies [6]. In addition, the rainbow connection number can also be motivated by its interesting interpretation in the area of networking [4]: Suppose that G represents a network, we wish to route messages between any two vertices in a pipeline, and require that each link on the route between the vertices (namely, each edge on the path) is assigned a distinct channel. Clearly, we want to minimize the number of distinct channels that we use in our network. This number is precisely rc(G).

The concept of rainbow connection number has several interesting variants, including strong rainbow connection number [5], [21], rainbow vertex-connection number [10], [15]. Uchizawa et al. [30] introduced another variant of rainbow connection, named rainbow total-connection (some researchers call it total rainbow connection, such as [22], [28], and we will use total rainbow connection in this paper). Let f be a total-coloring of $G=(V,E)$ which is not necessarily proper. A path P in G connecting two vertices u and v in V is called a total rainbow path between u and v if all elements in V(P) ∪ E(P), except u and v, are assigned distinct colors by f. The total-colored graph G is total rainbow connected if G has a total rainbow path between every two vertices in V. A graph G is called total rainbow k-connected if there is a total-coloring of G with k colors such that G is total rainbow connected. Now we define the totalrainbow connection number, denoted by trc(G), of a graph G as the minimum k such that it is total rainbow k-connected. Clearly, if H is a connected spanning subgraph of G, then trc(G) ≤ trc(H). There are more and more researchers investigating the topic of rainbow coloring, such as [1], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [17], [18], [20], [21], [22], [24], [25], [26], [27], [28], [29], [30]. The readers can see [16] for a survey and [19] for a monograph on it.

Uchizawa et al. [30] obtained some hardness and algorithmic results for total rainbow connection. It was shown in [30] that the problem that determine whether G is total rainbow connected is NP-Complete even for outerplanar graphs. In [25], Sun did some basic research on it and determined the precise values for total rainbow connection numbers of some special graph classes, including complete graphs, trees, cycles and wheels. Especially, it was shown in [25] that $trc\left(G\right)\le m\left(G\right)+n^{\prime }\left(G\right),$ and the equality holds if and only if G is a tree, where n′(G) is the number of inner vertices (that is, vertices of degrees at least two) of G. In [26], Sun showed that $trc\left(G\right)\ne m\left(G\right)+n^{\prime }\left(G\right)-1,m\left(G\right)+n^{\prime }\left(G\right)-2$ and characterized the graphs with $trc\left(G\right)=m\left(G\right)+n^{\prime }\left(G\right)-3$. With this result, the following sharp upper bound holds: for a connected graph G, if G is not a tree, then $trc\left(G\right)\le m\left(G\right)+n^{\prime }\left(G\right)-3$; moreover, the equality holds if and only if G belongs to five graph classes [26]. In the same paper, he also investigated the Nordhaus–Gaddum-type lower bounds for the rainbow total-connection number of a graph and derived that if G is a connected graph of order n ≥ 8, then $trc\left(G\right)+trc\left(\overline{G}\right)\ge 6$ and $trc\left(G\right)trc\left(\overline{G}\right)\ge 9$. An example is given to show that both of these bounds are sharp. In [28], Sun compared trc(G) with two other parameters of rainbow coloring, rc(G) and rvc(G). For an integer k ≥ 3, sufficient conditions that guarantee trc(G) ≤ k were determined. Among the results, he also got the sharp threshold function for a random graph to have trc(G) ≤ 3.

In this paper, we will continue to find bounds for the parameter trc(G). In Section 2, we will get a sharp upper bound for trc(G) in terms of the number of vertex-disjoint cycles in G (Theorem 2.1). Some special cases will also be discussed (Corollary 2.2).

In our paper, we will also study minimally total rainbow k-connected graphs, that is, total rainbow k-connected graphs with a minimum number of edges. For n, k ≥ 1, define h(n, k) to be the minimum size of a total rainbow k-connected graph G of order n. A network G which satisfies our requirements and has as few links as possible can reduce costs, shorten the construction period and simplify later maintenance. Thus, the study of this parameter is significant. In Section 3, we will compute exact values and upper bounds for h(n, k) (Theorem 3.10).