Sufficient conditions for graphs to be maximally 4-restricted edge connected

For a subset S of edges in a connected graph G, the set S is a k-restricted edge cut if G− S is disconnected and every component of G− S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. A connected graph G is said to be λk-connected if G has a k-restricted edge cut. Let ξk(G) = min{|[X, X̄ ]| : |X| = k, G[X] is connected}, where X̄ = V (G)\X. A graph G is said to be maximally k-restricted edge connected if λk(G) = ξk(G). In this paper we show that if G is a λ4-connected graph with λ4(G) ≤ ξ4(G) and the girth satisfies g(G) ≥ 8, and there do not exist six vertices u1, u2, u3, v1, v2 and v3 in G such that the distance d(ui, vj) ≥ 3, (1 ≤ i, j ≤ 3), then G is maximally 4-restricted edge connected. ∗ This work is supported by the National Natural Science Foundation of China (61772010). M. WANG ET AL. /AUSTRALAS. J. COMBIN. 70 (1) (2018), 123–136 124 1 Terminology and introduction We consider finite, undirected and simple graphs. For graph-theoretical terminology and notation not defined here we follow [5]. Let G be a graph with vertex set V = V (G) and edge set E = E(G). Given a nonempty vertex subset V ′ of V , the induced subgraph by V ′ in G, denoted by G[V ′], is a graph, whose vertex set is V ′ and the edge set is the set of all the edges of G with both endpoints in V ′. For two disjoint vertex sets X and Y of V , let [X, Y ] be the set of edges with one endpoint in X and the other one in Y . The order of G is the number of vertices in G. The degree of a vertex v in G, denoted by dG(v), is the number of edges of G incident with v. The set of neighbors of a vertex v in G is denoted by NG(v). A (v0, vk)-path, denoted by P = v0v1 . . . vk, is a sequence of adjacent vertices where all the vertices are distinct. Likewise, a cycle is a path that begins and ends with the same vertex. The length of a path or a cycle is the number of edges contained in the path or cycle. The distance between two vertices x and y is, denoted by d(x, y), the length of a shortest path between x and y in G. The girth g = g(G) is the length of a shortest cycle in G. Many multiprocessor systems have interconnection networks (networks for short) as underlying topologies and a network is usually represented by a graph where nodes represent processors and links represent communication links between processors. A classical measurement of the fault tolerance of a network is the edge connectivity λ(G). The edge connectivity λ(G) of a connected graph G is the minimum cardinality of an edge cut of G. As a more refined index than the edge connectivity, Fàbrega and Fiol [10] proposed the more general concept of the k-restricted edge connectivity of G as follows. Definition 1.1 [10] For a subset S of edges in a connected graph G, S is a krestricted edge cut if G − S is disconnected and every component of G − S has at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum k-restricted edge cut. A minimum k-restricted edge cut is called a λk-cut. A connected graph G is said to be λk-connected if G has a k-restricted edge cut. There is a significant amount of research on k-restricted edge connectivity [2, 4, 7– 11, 13, 18–21, 27]. In view of recent studies on k-restricted edge connectivity, it seems that the larger λk(G) is, the more reliable the network G is [3, 14, 22]. So, we expect λk(G) to be as large as possible. Clearly, the optimization of λk(G) requires an upper bound first and so the optimization of k-restricted edge connectivity draws a lot of attention. For any positive integer k, let ξk(G) = min{|[X, X̄ ]| : |X| = k, G[X] is connected}, where X̄ = V (G)\X. It has been shown that λk(G) ≤ ξk(G) holds for many graphs [1, 6, 12, 15, 28]. Let G1, . . . , Gn be n copies of Kt. Add a new vertex u and let u be adjacent to every vertex in V (Gi), i = 1, . . . , n. The resulting graph is denoted by G ∗ n,t. It can be verified that Gn,t has no (δ(G ∗ n,t) + 1)-restricted edge cuts and G ∗ n,t is the only exception for the existence of k-restricted edge cuts of a connected graph G when k ≤ δ(G) + 1. M. WANG ET AL. /AUSTRALAS. J. COMBIN. 70 (1) (2018), 123–136 125 Theorem 1.2 [28]. Let G be a connected graph with order at least 2(δ(G)+1) which is not isomorphic to any Gn,t with t = δ(G). Then for any k ≤ δ(G) + 1, G has k-restricted edge cuts and λk(G) ≤ ξk(G). A λk-connected graph G is said to be maximally k-restricted edge connected if λk(G) = ξk(G). When k = 2, the k-restricted edge connectivity of G is the restricted edge connectivity of G; a maximally k-restricted edge connected graph is a maximally restricted edge connected graph. There has been much research on maximally restricted edge connected graphs. See [13,17,22–24]. Let G be a λkconnected graph and let S be a λk-cut of G. In 1989, Plesńık and Znám [16] gave the following sufficient condition for a graph to be maximally edge connected. Theorem 1.3 [16] Let G be a connected graph. If there do not exist four vertices u1, u2, v1, v2 in G such that the distance d(ui, vj) ≥ 3 (1 ≤ i, j ≤ 2), then G is maximally edge connected. In 2013, Qin et al. [17] gave the following theorem. Theorem 1.4 [17] Let G be a λ2-connected graph with the girth g(G) ≥ 4. If there are not four vertices u1, u2, v1, v2 in G such that the distance d(ui, vj) ≥ 3 (1 ≤ i, j ≤ 2), then G is maximally restricted edge connected. In 2015, Wang et al. [25] gave the following theorem. Theorem 1.5 [25] Let G be a λ3-connected graph with the girth g(G) ≥ 5. If there are not five vertices u1, u2, v1, v2, , v3 in G such that the distance d(ui, vj) ≥ 3 (1 ≤ i ≤ 2; 1 ≤ j ≤ 3), then G is maximally 3-restricted edge connected. In this article, we extend the above result to λ4-connected graphs.