CCO Commun. Comb. Optim.

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1 Communications in Combinatorics and Optimization Vol. No. 1, 017 pp.35-1 DOI: 10.09/CCO CCO Commun. Comb. Optim. Sufficient conditions for maximally edge-connected and super-edge-connected graphs Lutz Volkmann 1 and Zhen-Mu Hong 1 Lehrstuhl II für Mathematik, RWTH Aachen University, 5056 Aachen, Germany volkm@math.rwth-aachen.de School of Finance, Anhui University of Finance and Economics, Bengbu, China zmhong@mail.ustc.edu.cn Received: 1 October 016; Accepted: 17 May 017; Available Online: 3 May 017. Communicated by Michitaka Furuya Abstract: Let G be a connected graph with minimum degree δ and edgeconnectivity λ. A graph is maximally edge-connected if λ δ, and it is superedge-connected if every minimum edge-cut is trivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree. In this paper, we show that a connected graph or a connected triangle-free graph is maximally edge-connected or super-edge-connected if the number of edges is large enough. Keywords: Edge-connectivity; Maximally edge-connected graphs; Superedge-connected graphs 010 Mathematics Subject Classification:05C0 1. Terminology and introduction Let G be a finite and simple graph with vertex set V V (G and edge set E E(G. The order and size of G are defined by n n(g V (G and m m(g E(G, respectively. If N(v N G (v is the neighborhood of This work was supported by the National Natural Science Foundation of China (No and the University Natural Science Research Project of Anhui Province (No. KJ016A003. Corresponding Author c 017 Azarbaijan Shahid Madani University. All rights reserved.

2 36 Sufficient conditions for maximally edge-connected... the vertex v V (G, then we denote by d(v N(v the degree of v and by δ δ(g the minimum degree of the graph G. For a subset X V (G, use G[X] to denote the subgraph of G induced by X. For two subsets X and Y of V (G let [X, Y ] be the set of edges with one endpoint in X and the other one in Y. We write K n for the complete graph of order n and K p,q for the complete bipartite graph whose partition sets have cardinality p and q, respectively. An edge-cut of a connected graph G is a set of edges whose removal disconnects G. The edge connectivity λ λ(g of a connected graph G is defined as the minimum cardinality of an edge-cut over all edge-cuts of G. An edge-cut S is a minimum edge-cut or a λ-cut if S λ(g. The inequality λ(g δ(g is immediate. We call a connected graph maximally edge-connected, if λ(g δ(g. In 1981, Bauer et al. [1] proposed the concept of super-edge connectedness. A graph is called super-edge-connected or superλ if every minimum edge-cut is trivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree. Thus every superedge-connected graph is also maximally edge-connected. Sufficient conditions for graphs to be maximally edge-connected or super-edgeconnected were given by several authors, see for example the survey paper by Hellwig and Volkmann [3]. The starting point was an article by Chartrand [] in He observed that if δ is large enough, then the graph is maximally edge-connected. A similar condition for super-edge-connectivity was given by Kelmans [] six years later. Over the years, these results have been strengthened many times and in many ways. However, there seems to be no result in the literature showing that graphs of sufficiently large size are maximally edgeconnected. In this paper we fill this gap. We show that a connected graph or a connected triangle-free graph G is maximally edge-connected or super-λ if the number of edges is large enough. For example, if m ( ( n δ 1 + δ+1 + δ 1, then G is maximally edge-connected, unless G is a graph obtained from K δ+1 K n δ 1 by adding δ 1 edges between K δ+1 and K n δ 1.. Maximally edge-connected graphs Theorem 1. Let G be a connected graph of order n, size m, minimum degree δ and edge-connectivity λ. If ( ( n δ 1 δ + 1 m + + δ 1, (1 then λ δ, unless G is a graph obtained from K δ+1 K n δ 1 by adding δ 1 edges

3 L. Volkmann and Zhen-Mu Hong 37 between K δ+1 and K n δ 1. Proof. Suppose to the contrary that λ δ 1. Let S be an arbitrary λ-cut, and let X and Y denote the vertex sets of the two components of G S with Y. Each vertex in X is adjacent to at most 1 vertices of X, and exactly λ edges join vertices of X to vertices of Y. Thus δ x X d(x ( 1 + λ, and so ( 1( δ δ λ > 0, which means that δ + 1. Thus Y δ + 1. Therefore we have δ + 1 Y n δ 1. Since there are no edges between X and Y in G S, we obtain m λ E(G S Y. This bound together with δ + 1 Y n δ 1, n and + Y n lead to m (δ + 1(n δ 1 + (δ 1 ( n δ 1 + ( δ δ 1. Combining this with (1, we have m ( ( n δ 1 + δ+1 + δ 1, and the two inequalities in the proof above must be equalities. Therefore λ δ 1, δ+1, Y n δ 1, G[X] K δ+1 and G[Y ] K n δ 1. This completes the proof. For the next result, we use the famous theorem of Mantel [5] and Turán [6]. Theorem. For any triangle-free graph G of order n, we have E(G 1 n, with equality if and only if G K n/, n/. Theorem 3. Let G be a connected triangle-free graph of order n, size m, minimum degree δ and edge-connectivity λ. If n m δ(n δ + δ 1, ( then λ δ, unless G is a triangle-free graph obtained from K δ,δ K n/ δ, n/ δ by adding δ 1 edges between K δ,δ and K n/ δ, n/ δ.

4 38 Sufficient conditions for maximally edge-connected... Proof. Suppose to the contrary that λ δ 1. Let S be an arbitrary λ-cut, and let X and Y denote the vertex sets of the two components of G S with Y. Using Theorem, we conclude that Y E(G[X] and E(G[Y ], (3 with equalities if and only if G[X] K /, / and G[Y ] K Y /, Y /. Note that E(G[X] x X d(x λ. Thus δ x X d(x + λ, and so δ + λ 0. It follows that ( 1( (δ 1 δ λ 1 1 > 0, which means that δ. Thus Y δ. Therefore we arrive at δ n Y n δ. This together with + Y n and (3 lead to m E(G[X] + E(G[Y ] + λ Y (n + + δ δ 1 (n + + δ ( n + δ 1 (as δ n n δ(n δ + δ 1 n δ(n δ + δ 1. Combining this with (, we have m δ(n δ+δ 1, and so λ δ 1, δ, Y n δ, E(G[X] δ and E(G[Y ] δn + δ. Therefore, G[X] K δ,δ, G[Y ] K n/ δ, n/ δ and [X, Y ] δ 1. This completes the proof.

5 L. Volkmann and Zhen-Mu Hong Super edge-connected graphs Theorem. Let G be a connected graph of order n, size m, minimum degree δ and edge-connectivity λ. If ( n m δ(n δ + δ, ( then G is super-λ, unless δ and G is a graph obtained from K δ K n δ by adding δ edges between K δ and K n δ such that the minimum degree of G is δ. Proof. Suppose to the contrary that G is not super-λ. Let S be an arbitrary λ-cut such that each of the two components of G S has at least two vertices. Let X and Y denote the vertex sets of the two components of G S with Y. Each vertex in X is adjacent to at most 1 vertices of X, and exactly λ edges join vertices of X to vertices of Y. Thus δ x X d(x ( 1 + λ ( 1 + δ, and so ( 1( δ 0, which means that δ. Thus Y δ. Therefore we have δ Y n δ. Since there are no edges between X and Y in G S, we obtain m λ E(G S Y. This bound together with δ Y n δ and + Y n lead to m δ(n δ + δ. Combining this with (, we have m ( n δ(n δ+δ, and the two inequalities in the proof above must be equalities. Therefore, λ δ, δ, Y n δ, G[X] K δ and G[Y ] K n δ. This completes the proof. Theorem 5. Let G be a connected triangle-free graph of order n, size m, minimum degree δ 3 and edge-connectivity λ. If (n + 1 m δ(n δ + 1, (5 then G is super-λ, unless G is a triangle-free graph obtained from K δ,δ 1 K n+1 n+1 δ, by adding δ edges between K δ δ,δ 1 and K n+1 that the minimum degree of G is δ. n+1 such δ, δ

6 0 Sufficient conditions for maximally edge-connected... Proof. Suppose to the contrary that G is not super-λ. Let S be an arbitrary λ-cut such that each of the two components of G S has at least two vertices. Let X and Y denote the vertex sets of the two components of G S with Y. Using Theorem, we conclude that Y E(G[X] and E(G[Y ], (6 with equalities if and only if G[X] K /, / and G[Y ] K Y /, Y /. Note that we have seen in the proof of Theorem 3 that δ x X d(x and so δ + δ 0, that is + λ ( 1( (δ 1 1. If, then δ. Nevertheless, the hypothesis is δ 3, which means 3, and so (δ (δ 1 1. Hence we have δ 1 and so Y δ 1. Therefore δ 1 n Y n δ + 1. This together with + Y n and (6 lead to + δ, m E(G[X] + E(G[Y ] + λ Y (n + + δ + + δ (n + + δ + 1 ( n + δ (as δ 1 n 1 (δ 1(n δ δ (n δ(n δ + 1 (n + 1 δ(n δ + 1. Combining this with (5, we have m (n+1 δ(n δ + 1, and so λ δ, δ 1, Y n δ + 1, E(G[X] δ(δ 1 and E(G[Y ] (n+1 δ(n δ. Therefore G[X] K δ,δ, G[Y ] K n+1 and [X, Y ] δ. δ, n+1 δ

7 L. Volkmann and Zhen-Mu Hong 1 Theorem 6. Let G be a connected triangle-free graph of order n, size m, minimum degree δ and edge-connectivity λ. If (n m + δ + 1, (7 then G is super-λ, unless G is a triangle-free graph obtained from K K n n, by adding δ edge(s between K and K n n such that the minimum degree of, G is δ. Proof. In the proof of Theorem 5, δ + δ 0. If δ, then is enough to guarantee δ + δ 0. Thus, with the same proceeding of the proof of Theorem 5, it is easy to get m + 1 ( n + λ (as n 1 (n + δ (n + δ + 1. Combining this with (7, we have m (n + δ + 1, and so λ δ,, Y n, E(G[X] 1 and E(G[Y ] (n. Therefore G[X] K, G[Y ] K n n, and [X, Y ] δ. References [1] D. Bauer, F. Boesch, C. Suffel, and R. Tindell, Connectivity extremal problems and the design of reliable probabilistic networks, The theory and application of graphs, Wiley, New York (1981, 5 5. [] G. Chartrand, A graph-theoretic approach to a communications problem, SIAM J. Appl. Math. 1 (1966, no., [3] A. Hellwig and L. Volkmann, Maximally edge-connected and vertexconnected graphs and digraphs: A survey, Discrete Math. 308 (008, no. 15, [] A.K. Kelmans, Asymptotic formulas for the probability of k-connectedness of random graphs, Theory Probab. Appl. 17 (197, no., 3 5. [5] W. Mantel, Problem 8, Wiskundige Opgaven 10 (1907, [6] P. Turán, On an extremal problem in graph theory (hungarian, Mat. Fiz. Lapok 8 (191, 36 5.