Maximal local edge-connectivity of diamond-free graphs

The edge-connectivity of a graph G can be defined as λ(G) = min{λG(u, v) |u, v ∈ V (G)}, where λG(u, v) is the local edge-connectivity of two vertices u and v in G. We call a graph G maximally edgeconnected when λ(G) = δ(G) and maximally local edge-connected when λG(u, v) = min{d(u), d(v)} for all pairs u and v of distinct vertices in G. In 2000, Fricke, Oellermann and Swart (unpublished manuscript) proved that a bipartite graph G of order n(G) is maximally local edgeconnected when n(G) ≤ 4δ(G) − 1. As an extension of this result, we will show in this work that it is sufficient for G to be diamond-free with n(G) ≤ 4δ(G)− 1 to guarantee the maximally local edge-connectivity. 1 Terminology and introduction We consider finite graphs without loops and multiple edges. The vertex set and edge set of a graph G are denoted by V (G) and E(G), respectively. For a vertex v ∈ V (G), the open neighborhood NG(v) = N(v) is the set of all vertices adjacent to v, and EG(v) = E(v) is the set of all edges incident with v. The numbers n(G) = |V (G)|, m(G) = |E(G)| and d(v) = |N(v)| are called the order, the size of G and the degree of v, respectively. The minimum degree of a graph G is denoted by δ(G) = δ. The edge-connectivity λ(G) of a graph G is the smallest number of edges whose deletion disconnects the graph. The local edge-connectivity λG(u, v) = λ(u, v) between two distinct vertices u and v of a graph G, is the maximum number of edgedisjoint u-v paths in G. It is a well-known consequence of Menger’s theorem [13] that λ(G) = min{λG(u, v) |u, v ∈ V (G)}. It is straightforward to verify that λ(G) ≤ δ(G) and λ(u, v) ≤ min{d(u), d(v)}. We call a graph G maximally edge-connected when λ(G) = δ(G) and maximally 154 ANDREAS HOLTKAMP local edge-connected when λ(u, v) = min{d(u), d(v)} for all pairs u and v of distinct vertices in G. The graph obtained from a complete graph of order 4 by removing an arbitrary edge is called a diamond. A graph G is then called diamond-free, if it contains no diamond as a (not necessarily induced) subgraph. Since λ(G) ≤ δ(G), there is a special interest in a graph G with λ(G) = δ(G). Different authors have presented sufficient conditions for a graph to be maximally edge-connected, as, for example Dankelmann and Volkmann [2, 3, 4], Fàbrega and Fiol [5], Fiol [6], Hellwig and Volkmann [8, 10], Lin, Miller and Rodger [12], Moriarty and Christopher [14], Volkmann [16, 18], and Wang, Xu and Wang [19]. For more information on this topic, we refer the reader to the survey article by Hellwig and Volkmann [11]. However, closely related investigations for the local edge-connectivity have received little attention until recently. Fricke, Oellermann and Swart [7] studied the local edge-connectivity of p-partite graphs and graphs with bounded diameter. Hellwig and Volkmann [9] and Volkmann [17] gave sufficient conditions for the maximally local edge-connectivity of p-partite digraphs and graphs with bounded clique number. In [15], Volkmann proved that bipartite graphs G with n(G) ≤ 4δ(G) − 1 are maximally edge-connected. Fricke, Oellermann and Swart [7] showed that this condition even guarantees the maximally local edge-connectivity of G. By looking at diamond-free graphs Dankelmann et al. [1] were able to generalize a similar result on the maximally local (vertex-)connectivity of bipartite graphs. In this work, we will give a generalization of the results of Volkmann and Fricke et al. by proving that it is sufficient for G to be diamond-free with n(G) ≤ 4δ(G)− 1 to imply maximally local edge-connectivity. 2 Main Result Theorem 2.1: Let G be a diamond-free graph with δ(G) ≥ 3. If n(G) ≤ 4δ(G)− 1, then G is maximally local edge-connected. Proof: Assume G is not maximally local edge-connected. Therefore, we have two vertices u, v ∈ V (G) with r = min{d(u), d(v)} − δ ≥ 0 and an edge set S separating u and v with |S| ≤ δ + r − 1. Let U be the component of G − S with u ∈ V (U). Since n ≤ 4δ − 1 and by symmetry of u and v, without loss of generality, we may assume n(U) ≤ 2δ − 1. (1) Furthermore, since d(u) ≥ δ+ r > |S| the vertex u must have at least one neighbour in V (U) and, in addition, at least for one neighbour u′ ∈ V (U) of u, we have E(u′) ∩ S = ∅ (i.e. none of the edges incident with u′ is in S). We distinguish two cases: Case 1. u and u′ have a common neighbour in V (U). MAXIMAL LOCAL EDGE-CONNECTIVITY 155 Let u′′ ∈ N(u) ∩ N(u′) ∩ V (U). Since G is diamond-free, u, u′ and u′′ can have no further common neighbours (pairwise). Let W = (N(u) ∩ V (U)) \ {u′, u′′},W ′ = (N(u′) ∩ V (U))) \ {u, u′′} and W ′′ = (N(u′′) ∩ V (U)) \ {u, u′}. Since G is diamondfree, W ∩W ′ = ∅,W ∩W ′′ = ∅ and W ′ ∩W ′′ = ∅. By T = E(u) ∩ S we refer to the edges of S incident with u, and let T ′ = E(u′)∩S and T ′′ = E(u′′)∩S, respectively. Since no edge incident with u′ is in S, we have |W ′| ≥ δ − 2. (2) Together with (1) this leads to 2δ − 1 ≥ n(U) ≥ |W |+ |W ′|+ |W ′′|+ 3 ≥ |W |+ |W ′′|+ δ + 1. Hence we have |W |+ |W ′′| ≤ δ − 2. (3) Obviously, we have |T |+ |W | ≥ δ+ r− 2 and |T ′′|+ |W ′′| ≥ δ− 2. Thus, we deduce 2δ + r − 4 ≤ |T |+ |T ′′|+ |W |+ |W ′′| (3) ≤ |T |+ |T ′′|+ δ − 2, which implies |T |+ |T ′′| ≥ δ + r − 2. (4) We now take a closer look at the vertices in W ′. Assume there is a vertex w ∈ W ′ with E(w) ∩ S = ∅. Since G is diamond-free, w cannot be adjacent to u or u′′, and have at most one neighbour in W ′. Therefore, it follows that 2δ − 1 (1) ≥ n(U) ≥ |N(w) \ (W ′ ∪ {u′})| + |W ′|+ |{u, u′, u′′}| (2) ≥ δ − 2 + δ − 2 + 3 = 2δ − 1. So w must have exactly one neighbour w′ ∈ W ′ which cannot have further neighbours in U , and, of course, δ ≥ 4. Since G is diamond-free, w′ is only adjacent to w and u′, but cannot have neighbours in (N(w)\{u′, w′})∪{u, u′′}∪(W ′\{w}). Thus, w′ must have at least δ−2 incident edges in S, i.e. |E(w′)∩S| ≥ δ−2. Hence, every vertex in W ′ is either incident with at least one edge in S, or has exactly one neighbour in W ′ with at least 2 incident edges in S, and this neighbour cannot have further neighbours in W ′. As a consequence, with δ ≥ 4 and |T ′| = |E(W ′)∩ S| = |{E(x)|x ∈ W ′} ∩ S| we obtain |T ′| ≥ |W ′| ≥ δ − 2. (5) By combining (5) with (4), we now deduce |S| ≥ |T |+ |T ′|+ |T ′′| ≥ δ + r − 2 + δ − 2 = δ + r + (δ − 4) which is a contradiction to |S| ≤ δ + r − 1 for δ ≥ 4. In case δ = 3 this deduction shows that all edges in S are incident with either u, u′′ or w, where w ∈ W ′, and 156 ANDREAS HOLTKAMP since |S| = r + 2, |T | + |T ′′| = r + 1 and |T ′| = |W ′| = 1, the vertex w must have exactly one incident edge in S and one more neighbour x ∈ V (U) besides u′. Since G is diamond-free, x can now only be adjacent to at most one of the vertices u, u′ and u′′. Hence, x must either have one more neighbour in U leading to n(U) ≥ 6 = 2δ, or an edge of S must be incident with x, a contradiction on the size of U or S. Case 2. u and u′ have no common neighbour in V (U). Again, we define W = (N(u) ∩ V (U)) \ {u′} and W ′ = (N(u′) ∩ V (U)) \ {u}. Now W ∩W ′ = ∅ and |W ′| ≥ δ − 1. (6) Let T = E(u) ∩ S. Since |W | ≥ δ + r − 1− |T |, we conclude 2δ − 1 (1) ≥ n(U) ≥ |W |+ |W ′|+ 2 ≥ δ + r − 1− |T |+ δ − 1 + 2 = 2δ + r − |T | and, therefore, |T | ≥ r + 1. (7) Here (6) and (7) together with |S| ≤ δ+r−1 lead to the conclusion that there must be a vertex u′′ ∈ W ′ such that no edge in S is incident with u′′. Now u′′ can have at most one neighbour in W ′, hence we have |W ′′| ≥ δ−2 where W ′′ = N(u′′)\(W ′∪{u, u′}), which leads us to 2δ − 1 (1) ≥ n(U) ≥ |W ′|+ |W ′′|+ 2 ≥ δ − 1 + δ − 2 + 2 = 2δ − 1. We conclude that u′′ must have a neighbour w′ ∈ W ′, and since G is diamond-free, w′ must be incident with at least δ−2 edges in S, i.e. |T ′| ≥ δ−2 where T ′ = E(w′)∩S. Furthermore, we must have W ⊆ W ′′, and δ + r − 1 ≥ |S| ≥ |T |+ |T ′| (7) ≥ r + 1 + δ − 2 = δ + r − 1. Then it follows that |T | = r + 1 and thus |W | = δ − 2 and W = W ′′. Now, an arbitrary vertex w ∈ W (= W ′′) cannot be adjacent to u′ or w′, and no edge in S is incident with w. Furthermore, w cannot have a neighbour in W , otherwise we would have a diamond in U together with u and u′′. Thus, W ∩ (N(w)\ {u, u′′} = ∅, leading us to 2δ − 1 (1) ≥ n(U) ≥ |W |+ |N(w) \ {u, u′′}|+ |{u, u′, u′′, w′}| ≥ δ − 2 + δ − 2 + 4 = 2δ, a contradiction. As a direct consequence of Theorem 2.1 we obtain the following results. Corollary 2.2: Let G be a diamond-free graph with δ(G) ≥ 3. If n(G) ≤ 4δ(G)−1, then G is maximally edge-connected. Corollary 2.3 (Volkmann [15]) Let G be a bipartite graph with δ(G) ≥ 3. If n(G) ≤ 4δ(G)− 1, then G is maximally edge-connected. MAXIMAL LOCAL EDGE-CONNECTIVITY 157 Corollary 2.4 (Fricke, Oellermann, Swart [7]) Let G be a bipartite graph with δ(G) ≥ 3. If n(G) ≤ 4δ(G)− 1, then G is maximally local edge-connected. To see that Theorem 2.1 and Corollary 2.2 are sharp in the sense that for every integer p there exists a diamond-free graph G with δ(G) = p and n(G) = 4δ(G), which is not maximally edge-connected and, therefore, not maximally local edgeconnected, we consider the following example. Example 2.5: Let G be the graph obtained from two complete bipartite graphs Kp,p (p ≥ 2) by adding one arbitrary edge between them. Of course, G is diamondfree with edge-connectivity λ(G) = 1, while δ(G) = p and n(G) = 4δ(G). Therefore, G is not maximally edge-connected and not maximally local edge-connected. To see that Theorem 2.1 does not hold for δ(G) = 2, we consider the following graph. Example 2.6: Let G be the graph obtained from two 3-cycles by adding one arbitrary edge between them. Then δ(G) = 2, but λ(G) = 1. Thus, G is not maximally edge-connected and not maximally local edge-connected, but we have n(G) = 6 ≤ 7 = 4δ(G)− 1.