Sufficient conditions for maximally edge-connected graphs and arc-connected digraphs

A connected graph with edge connectivity λ and minimum degree δ is called maximally edge-connected if λ = δ. A strongly connected digraph with arc connectivity λ and minimum degree δ is called maximally arc-connected if λ = δ. In this paper, some new sufficient conditions are presented for maximally edge-connected graphs and arc-connected digraphs. We only consider finite graphs (digraphs) without loops and multiple edges (arcs). Let D be a digraph with vertex set V (D) and arc set A(D). For a vertex v of D, denote N(v) = {u ∈ V (D) : vu ∈ A(D)}, N−(v) = {w ∈ V (D) : wv ∈ A(D)}, d(v) = |N+(v)|, d−(v) = |N−(v)| and δ = δ(D) = min{d+(v), d−(v) : v ∈ V (D)}. For uv ∈ A(D), uv is called isolated, if N(u) = {v}, N−(u) = ∅, N(v) = ∅ and N−(v) = {u}. For two disjoint vertex sets X and Y of D, let [X, Y ] be the set of arcs from X to Y and D[X] be the subdigraph induced by X. A digraph is strongly connected if for every pair u, v of vertices there exists a directed path from u to v in D. For a strongly connected digraph D, a set of arcs W ⊆ A(D) is an arc cut if D − W is not strongly connected. A k-arc cut is an arc cut of order k and the arc connectivity λ = λ(D) of D is the minimum value of k. D is called maximally arc-connected if λ = δ. For Similar definitions and notations for graphs we refer the reader to [2]. The study of maximally edge-connected graphs and arc-connected digraphs has special relevance to the design of reliable networks. This is due to the fact that the higher the edge connectivity, the more reliable the network. Sufficient conditions for ∗ This work is supported by the National Natural Science Foundation of China (61070229). † Corresponding author. 234 SHIYING WANG, GUOZHEN ZHANG AND XIULI WANG maximally edge-connected graphs and arc-connected digraphs were given by several authors. LetG be a connected graph with |V (G)| = n. Chartrand [3] gave a sufficient condition δ ≥ n/2 for λ = δ. Lesniak [12] weakened the condition δ ≥ n/2 to d(u) + d(v) > n for all pairs of nonadjacent vertices u and v in G. Goldsmith and White [8] proved that it is sufficient to have n/2 disjoint pairs of vertices ui, vi with d(ui) + d(vi) ≥ n. Bollobás [1] gave a degree sequence condition that includes the condition of Goldsmith and White for odd n. Xu [17] gave an analogue for digraphs to the result by Goldsmith and White. Dankelmann and Volkmann [4] generalized Bollobás’ and Xu’s results and gave analogous degree conditions for bipartite graphs. The more conditions for λ = δ were given in [5, 6, 7, 9, 10, 11, 13, 14, 15, 16]. Let u and v be two vertices of a graph (digraph) D. The distance dD(u, v) = d(u, v) from u to v is the length of a shortest (directed) path from u to v in D. Let X and Y be two vertex sets of D. The distance dD(X, Y ) from X to Y is given by dD(X, Y ) = d(X, Y ) = min{d(x, y) : x ∈ X, y ∈ Y }. A pair of vertex sets X and Y of D with distance dD(X, Y ) = k is called k-distance maximal, if there exist no vertex sets X1 ⊇ X and Y1 ⊇ Y with X1 = X or Y1 = Y such that dD(X1, Y1) = k. Let D be a bipartite graph (digraph) with the bipartition V (D) = V ′ ∪ V ′′. For X ⊆ V (D), denote X ′ = X ∩ V ′ and X ′′ = X ∩ V ′′. A pair of vertex sets X and Y of D with dD(X ′, Y ′) = k and dD(X ′′, Y ′′) = k is called (k, k)-distance maximal, if there exist no vertex sets X1 ⊇ X and Y1 ⊇ Y with X1 = X or Y1 = Y such that dD(X ′ 1, Y ′ 1) = dD(X ′′ 1 , Y ′′ 1 ) = k. In 2003, Hellwig and Volkmann presented sufficient conditions for maximally arc-connected digraphs in term of the isolated vertex in [10]. Theorem 1 ([10]). Let D be a strongly connected digraph with arc connectivity λ and minimum degree δ. If for all 3-distance maximal pairs of vertex sets X and Y there exists an isolated vertex in D[X ∪ Y ], then λ = δ. Theorem 2 ([10]). Let D be a strongly connected bipartite digraph with arc connectivity λ and minimum degree δ. If for all (4,4)-distance maximal pairs of vertex sets X and Y there exists an isolated vertex in D[X ∪ Y ], then λ = δ. In this paper, inspired by Theorems 1 and 2, we present some new sufficient conditions for maximally edge-connected graphs and arc-connected digraphs in term of the isolated arc. Let D be a strongly connected digraph with arc connectivity λ. By the definition of the arc connectivity λ, there exists an arc cut [S, T ] with |[S, T ]| = λ, where S ⊆ V (D) and T = V (D)\S. Let A ⊆ S and B ⊆ T be the sets of vertices incident with at least an arc of [S, T ]. Then |A|, |B| λ. Define A0 = S\A and B0 = T\B. In the following, we give an important lemma. Lemma 1. If there exist two disjoint, nonempty sets X, Y ⊆ V (D) such that A0 ⊆ X,B0 ⊆ Y and there exists an isolated arc uv in D[X ∪ Y ], then λ = δ. Proof. Suppose, to the contrary, that λ < δ. We investigate four cases. MAXIMALLY EDGE-CONNECTED GRAPHS 235 Case 1. u ∈ A0. By the definition of A0, we have v ∈ A0 or v ∈ A. If v ∈ A0, then, by the definitions of the isolated arc and A0, we obtain N (u) ⊆ A ∪ {v} and N(v) ⊆ A. Therefore δ > |A| |(N+(u) ∪N+(v))\{v}| = |N+(u)\{v}|+ |N+(v)| − |N+(u) ∩N+(v)| d(u) − 1 + d+(v)−min{d+(u)− 1, d+(v)} = max{d+(u) − 1, d+(v)} δ, a contradiction. If v ∈ A, then, by the definitions of the isolated arc, A0 and A, we obtain N(u) ⊆ A and N(v) ⊆ A ∪B. Therefore 2δ d(u) + d(v) = |N+(u) ∩A|+ |N+(v) ∩ B|+ |N+(v) ∩ A| |A|+ |N+(v) ∩ B|+ ∑ x∈N+(v)∩A |N+(x) ∩ B| |A|+ ∑ x∈A |N+(x) ∩ B| < δ + λ < 2δ, a contradiction. Case 2. u ∈ A. By the definition of A, we have v ∈ A0, v ∈ A, or v ∈ B. If v ∈ A0, then, by the definitions of the isolated arc, A0 and A, we obtain N (u) ⊆ A ∪ B ∪ {v} and N(v) ⊆ A\{u}. Therefore 2δ d(u) + d(v) = |N+(u) ∩B| + |N+(u) ∩ A|+ 1 + |N+(v) ∩ A\{u}| |N+(u) ∩B| + ⎛ ⎝ ∑ x∈N+(u)∩A |N+(x) ∩ B| ⎞ ⎠ + 1 + |A| − 1 (∑ x∈A |N+(x) ∩ B| ) + |A| < λ+ δ < 2δ, a contradiction. If v ∈ A, then, by the definitions of the isolated arc and A, we obtain N(u) ⊆ A ∪ B and N(v) ⊆ A ∪ B. Therefore 2δ d(u) + d(v) = |N+(u) ∩ B|+ |N+(u) ∩ A|+ |N+(v) ∩ B|+ |N+(v) ∩ A| |N+(u) ∩ B|+ ⎛ ⎝ ∑ x∈N+(u)∩A |N+(x) ∩ B| ⎞ ⎠ 236 SHIYING WANG, GUOZHEN ZHANG AND XIULI WANG