On local connectivity of K2, p-free graphs

For a vertex v of a graph G, we denote by d(v) the degree of v. The local connectivity κ(u, v) of two vertices u and v in a graph G is the maximum number of internally disjoint u–v paths in G. Clearly, κ(u, v) ≤ min{d(u), d(v)} for all pairs u and v of vertices in G. We call a graph G maximally local connected when κ(u, v) = min{d(u), d(v)} for all pairs u and v of distinct vertices in G. Let p ≥ 2 be an integer. We call a graph K2,p-free if it contains no complete bipartite graph K2,p as a (not necessarily induced) subgraph. If p ≥ 3 and G is a connected K2,p-free graph of order n and minimum degree δ such that n ≤ 3δ−2p+2, then G is maximally local connected due to our earlier result on p-diamond-free graphs [Discrete Math. 309 (2009), 6065–6069]. Now we present examples showing that the condition n ≤ 3δ− 2p+2 is best possible for p = 3 and p ≥ 5. In the case p = 4 we present the improved condition n ≤ 3δ − 5 implying maximally local connectivity. In addition, we present similar results for K2,2-free graphs. 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 30 ANDREAS HOLTKAMP AND LUTZ VOLKMANN v ∈ V (G), the open neighborhood NG(v) = N(v) is the set of all vertices adjacent to v, and NG[v] = N [v] = N(v) ∪ {v} is the closed neighborhood of v. If A ⊆ V (G), then NG[A] = ⋃ v∈A NG[v], and G[A] is the subgraph induced by A. The numbers |V (G)| = n(G) = n, |E(G)| = m(G) = m and |N(v)| = dG(v) = d(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. For an integer p ≥ 2, we define a p-diamond as the graph with p + 2 vertices, where two adjacent vertices have exactly p common neighbors, and the graph contains no further edges. For p = 2, the 2-diamond is the usual diamond. A graph is p-diamond-free if it contains no p-diamond as a (not necessarily induced) subgraph. The complete graph of order n is denoted by Kn. Let Ks,t be the complete bipartite graph with the bipartition A,B such that |A| = s and |B| = t. We call a graph Ks,t-free if it contains no Ks,t as a (not necessarily induced) subgraph. Notice that in the special case s = t = 2, the graph K2,2 is isomorphic to the cycle C4 of length 4. The connectivity κ(G) of a connected graph G is the smallest number of vertices whose deletion disconnects the graph or produces the trivial graph (the latter only applying to complete graphs). The local connectivity κG(u, v) = κ(u, v) between two distinct vertices u and v of a connected graphG, is the maximum number of internally disjoint u–v paths in G. It is a well-known consequence of Menger’s theorem [11] that κ(G) = min{κG(u, v) |u, v ∈ V (G)}. (1) It is straightforward to verify that κ(G) ≤ δ(G) and κ(u, v) ≤ min{d(u), d(v)}. We call a graph G maximally connected when κ(G) = δ(G) and maximally local connected when κ(u, v) = min{d(u), d(v)} for all pairs u and v of distinct vertices in G. Because of κ(G) ≤ δ(G), there exists a special interest on graphs G with κ(G) = δ(G). Different authors have presented sufficient conditions for graphs to be maximally connected, as, for example Balbuena, Cera, Diánez, Garćıa-Vázquez and Marcote [1], Esfahanian [3], Fàbrega and Fiol [4, 5], Fiol [7], Hellwig and Volkmann [8], Soneoka, Nakada, Imase and Peyrat [12] and Topp and Volkmann [13]. For more information on this topic we refer the reader to the survey articles by Hellwig and Volkmann [9] and Fàbrega and Fiol [6]. However, closely related investigations for the local connectivity have received little attention until recently. In this paper we will present such results for K2,p-free graphs. We start with a simple and well-known proposition. Observation 1 If a graph G is maximally local connected, then it is maximally connected. Proof. Since G is maximally local connected, we have κ(u, v) = min{d(u), d(v)} for all pairs u and v of vertices in G. Thus (1) implies κ(G) = min u,v∈V (G) {κ(u, v)} = min u,v∈V (G) {min{d(u), d(v)}} = δ(G). LOCAL CONNECTIVITY OF K2,p-FREE GRAPHS 31 2 K2,p-free graphs with p ≥ 3 Recently, Holtkamp and Volkmann [10] gave a sufficient condition for connected p-diamond-free graphs to be maximally local connected. Theorem 2 (Holtkamp and Volkmann [10] 2009) Let p ≥ 3 be an integer, and let G be a connected p-diamond-free graph. If n(G) ≤ 3δ(G) − 2p + 2, then G is maximally local connected. Since a K2,p-free graph is also p-diamond-free, the next corollary is immediate. Corollary 3 Let p ≥ 3 be an integer, and let G be a connected K2,p-free graph. If n(G) ≤ 3δ(G)− 2p+ 2, then G is maximally local connected. The next result is a direct consequence of Corollary 3 and Observation 1. Corollary 4 Let p ≥ 3 be an integer, and let G be a connected K2,p-free graph. If n(G) ≤ 3δ(G)− 2p+ 2, then G is maximally connected. The following examples will demonstrate that the condition n(G) ≤ 3δ(G)−2p+2 in Corollaries 3 and 4 is best possible for p = 3 and p ≥ 5. Example 5 The connected graph in Figure 1 isK2,3-free with minimum degree δ = 4 and order n = 3δ − 6 + 3 = 9. The vertex set S with |S| = 3 disconnects the graph, and therefore it is neither maximally connected nor maximally local connected. Thus the condition n(G) ≤ 3δ(G)−2p+2 in Corollaries 3 and 4 are best possible for p = 3. Figure 1: K2,3-free graph with δ = 4 and n = 3δ − 3 = 9 vertices which is not maximally (local) connected. Let G3, G4, G5 and G6 be the graphs depicted in Figure 2. Each Gp is a connected K2,p-free graph with δ(Gp) = p and n(Gp) = 3δ(Gp)−2p+3 = p+3. The graphs G5 and G6 are not maximally connected and therefore not maximally local connected, since the removal of the vertex set S with |S| = δ(Gp) − 1 = p − 1 disconnects the graphs. So Corollaries 3 and 4 are best possible for p = 5 and p = 6. 32 ANDREAS HOLTKAMP AND LUTZ VOLKMANN Figure 2: K2,p-free graphs Gp (p ∈ {3, 4, 5, 6}) with δ(Gp) = p and n = 3δ(Gp)−2p+ 3 = δ(Gp) + 3 = p+ 3. The graphs G5 and G6 are not maximally (local) connected, G3 and G4 are. Starting with the four graphs G3, G4, G5 and G6, we are able to construct successively similar graphs Gp for all p ≥ 7. Each Gp will be connected and K2,p-free with δ(Gp) = p and n(Gp) = 3δ(Gp)− 2p+3 = p+3. A vertex set S with |S| = p− 1 will separate Gp, showing that neither of the graphs is maximally connected or maximally local connected. Given a graph Gp with the described properties, we can construct a graph Gp+4 with the same qualities in the subsequently specified way. For Gp+4 not to be maximally (local) connected the maximally (local) connectivity of Gp is irrelevant (e.g. G3 and G4 are maximally (local) connected). The existence of Gp for all p ≥ 7 then follows by induction. So let Gp be a graph with the properties mentioned above. We obtain the graph Gp+4 by adding four new vertices u, u ′, v and v′, the edges uu′ and vv′ as well as all possible edges between the four new vertices and the vertices of Gp that means {xy|x ∈ {u, u′, v, v′} and y ∈ V (Gp)}. Then n(Gp+4) = n(Gp) + 4 = p + 3 + 4 = (p + 4) + 3 and δ(Gp+4) = δ(Gp) + 4 = n(Gp) + 1 = p + 4. We will now show that Gp+4 is K2,p+4-free. So let w and z be two arbitrary vertices of Gp+4. We distinguish three different cases. Case 1. Assume that w, z ∈ {u, u′, v, v′}. Then w and z can only have common neighbors in Gp. Because n(Gp) = p + 3, the vertices w and z have at most p + 3 common neighbors. Case 2. Assume that w ∈ {u, u′, v, v′} and z ∈ V (Gp). Without loss of generality, LOCAL CONNECTIVITY OF K2,p-FREE GRAPHS 33 we can assume that w = u. Therefore w and z only have |{u′}∪(V (Gi)−{z})| = p+3 common neighbors. Case 3. Assume that w, z ∈ V (Gp). Since Gp is K2,p-free, w and z again have at most (p − 1) + 4 = p+ 3 common neighbors. We have seen that no two vertices in Gp+4 could have more than p+ 3 common neighbors. Therefore Gp+4 is K2,p+4-free. Since Gp+4 − V (Gp) is disconnected with n(Gp) = p+3 and δ(Gp+4) = p+4, the graph Gp+4 is not maximally connected and therefore not maximally local connected. Next we will present an improved condition on maximally local connectivity for K2,4-free graphs. For the proof we use the following result. Theorem 6 (Holtkamp and Volkmann [10] 2009) Let p ≥ 2 be an integer, and let G be a connected p-diamond-free graph. In addition, let u, v ∈ V (G) be two vertices of G and define r = min{dG(u), dG(v)} − δ(G). (1) If uv ∈ E(G) and n(G) ≤ 3δ(G) + r − 2p+ 2, then κG(u, v) = δ(G) + r. (2) If uv ∈ E(G) and n(G) ≤ 3δ(G) + r − 2p+ 1, then κG(u, v) = δ(G) + r. Theorem 7 Let G be a connected K2,4-free graph with minimum degree δ(G) ≥ 3. If n(G) ≤ 3δ(G)− 5, then G is maximally local connected. Proof. If n(G) ≤ 3δ(G) − 6, then the maximally local connectivity of G follows from Corollary 3. Thus let now n(G) = 3δ(G) − 5. If δ(G) = 3, then n(G) = 4 and therefore G is isomorphic to the complete graph K4, which is maximally local connected. In the case δ(G) ≥ 4, we suppose to the contrary that G is not maximally local connected. This means that there are two vertices u, v ∈ V (G) with κG(u, v) ≤ δ(G) + r − 1 for r = min{dG(u), dG(v)} − δ(G). Next we distinguish two cases. Case 1. Assume that uv ∈ E(G). As a K2,4-free graph is also 4-diamond-free, Theorem 6(2) implies 0 ≤ r ≤ 1. If we define the graph H by H = G − uv, then there exists a vertex set S ⊂ V (H) = V (G) with |S| ≤ δ(G) + r − 2 that separates u and v in H. Because dH(u) ≥ δ + r − 1 and dH(v) ≥ δ + r − 1, there is a vertex u′ ∈ V (H) − S adjacent to u as well as a vertex v′ ∈ V (H) − S adjacent to v in H. Since H is also K2,4-free, we deduce that |NH [{u, u′}]| ≥ 2δ(G) + r − 4 as well as |NH [{v, v′}]| ≥ 2δ(G)+ r− 4. Combining these two bounds with |S| ≤ δ(G)+ r− 2, we obtain n(G) = 3δ(G)− 5 ≥ |NH [{u, u′}]|+ |NH [{v, v′}]| − |S| ≥ 4δ(G) + 2r − 8− |S| ≥ 4δ(G) + 2r − 8− (δ(G) + r − 2) = 3δ(G) + r − 6. 34 ANDREAS HOLTKAMP AND LUTZ VOLKMANN In view of 0 ≤ r ≤ 1, this inequality chain shows that H − S consists of exactly two components with vertex sets Wu and Wv such that u ∈ Wu and v ∈ Wv. In addition, the inequality 3δ(G)− 5 ≥ 4δ(G) + 2r − 8− |S| leads to |S| = δ(G)− 1 when r = 1 and δ(G)− 3 ≤ |S| ≤ δ(G)− 2 when r = 0. Subcase 1.1. Assume that r = 1. Then |S| = δ(G) − 1 and therefore |Wu| = |Wv| = δ(G)− 2. Subcase 1.1.1. Assume that δ(G) = 4. Then |S| = 3, Wu = {u, u′} and Wv = {v, v′}. Because δ(H) ≥ 4, each vertex of {u, u′, v, v′} is adjacent to each vertex in S. Hence G contains a K2,4 as a subgraph, a contradiction to the hypothesis. Subcase 1.1.2. Assume that δ(G) = 5. Then |S| = 4 and |Wu| = |Wv| = 3. Because δ(H) ≥ 5, each vertex of Wu ∪Wv is adjacent to at least three vertices in S. Hence there exist at least two vertices w and z in S such that w has 6 neighbors in Wu ∪ Wv and z has 4 neighbors in Wu ∪ Wv or w has 5 neighbors in Wu ∪ Wv and z has 5 neighbors in Wu ∪Wv. In both cases G contains a K2,4 as a subgraph, a contradiction. Subcase 1.1.3. Assume that δ(G) = 6. Then |S| = 5 and |Wu| = |Wv| = 4. Assume first that Wu contains a vertex w adjacent to all vertices in S. If there exists a vertex w′ ∈ Wu − {w} with 4 neighbors in S, then G contains a K2,4 as a subgraph, a contradiction. If each vertex in Wu − {w} has at most 3 neighbors in S, then G[Wu] is isomorphic to the complete graph K4. Now an arbitrary vertex w′ ∈ Wu − {w} and w have two common neighbors in Wu and at least 3 common neighbors in S, a contradiction. Assume secondly that each vertex of Wu has at most 4 neighbors in S. Then G[Wu] is either a cycle C4, a diamond or a K4. In the first two cases there are two vertices w and z in Wu sharing two neighbors in Wu and at least 3 in S, a contradiction. In the last case every two vertices in Wu have two common neighbors in Wu, and since every vertex of Wu has at least 3 neighbors in S, it is easy to see that G contains a K2,4 as a subgraph, a contradiction. Subcase 1.1.4. Assume that δ(G) ≥ 7. Then |Wu| ≥ 5. Let w1, w2, w3 ∈ Wu be three pairwise distinct vertices. Since G is K2,4-free and δ(H) = δ(G), it is straightforward to verify that |NH [{w1, w2, w3}]| ≥ 3δ(G)− 9. We deduce that 3δ(G)− 5 = n(G) ≥ |NH [{w1, w2, w3}]|+ |Wv| ≥ 4δ(G)− 11, and we obtain the contradiction δ(G) ≤ 6. Subcase 1.2. Assume that r = 0 and |S| = δ(G)−3. Then |Wu| = |Wv| = δ(G)−1. Subcase 1.2.1. Assume that δ(G) = 4. Then |S| = 1 and |Wu| = 3. However, this is impossible, since dH(u ′) ≥ 4 for u′ ∈ (Wu − {u}). Subcase 1.2.2. Assume that δ(G) = 5. Then |S| = 2 and |Wu| = |Wv| = 4. Hence every vertex in (Wu ∪Wv) − {u, v} is adjacent to every vertex in S. So G contains a K2,4 as a subgraph, a contradiction. LOCAL CONNECTIVITY OF K2,p-FREE GRAPHS 35 Subcase 1.2.3. Assume that δ(G) ≥ 6. Then |Wu| ≥ 5. Let w1, w2, w3 ∈ (Wu − {u}) be three pairwise distinct vertices. Since G is K2,4-free and dH(wi) ≥ δ(G) ≥ 6 for 1 ≤ i ≤ 3, we conclude that |NH [{w1, w2, w3}]| ≥ 3δ(G) − 9. This yields the contradiction 3δ(G)− 5 = n(G) ≥ |NH [{w1, w2, w3}]|+ |Wv| ≥ 4δ(G)− 10 ≥ 3δ(G)− 4. Subcase 1.3. Assume that r = 0 and |S| = δ(G) − 2. Then, without loss of generality, |Wu| = δ(G)− 2 and |Wv| = δ(G)− 1. Subcase 1.3.1. Assume that δ(G) = 4. Then |S| = 2 and |Wu| = 2. However, this is impossible, since dH(u ′) ≥ 4 for u′ ∈ (Wu − {u}). Subcase 1.3.2. Assume that δ(G) = 5. Then |S| = |Wu| = 3. If Wu = {u, u′, u′′}, then u′ as well as u′′ is adjacent to every vertex in S ∪ {u}. So G contains a K2,4 as a subgraph, a contradiction. Subcase 1.2.3. Assume that δ(G) ≥ 6. Then |Wu| ≥ 4. Let w1, w2, w3 ∈ (Wu − {u}) be three pairwise distinct vertices. Since G is K2,4-free and dH(wi) ≥ δ(G) ≥ 6 for 1 ≤ i ≤ 3, it follows that |NH [{w1, w2, w3}]| ≥ 3δ(G) − 9. Therefore we obtain the contradiction 3δ(G)− 5 = n(G) ≥ |NH [{w1, w2, w3}]|+ |Wv| ≥ 4δ(G)− 10 ≥ 3δ(G)− 4. Case 2. Assume that uv ∈ E(G). Now Theorem 6(1) implies r = 0. So there exists a vertex set S ⊂ V (G) with |S| ≤ δ(G)−1 that separates u and v in G. Hence there is a vertex u′ ∈ V (G) − S adjacent to u as well as a vertex v′ ∈ V (G) − S adjacent to v. Since G is K2,4-free, we deduce that |NG[{u, u′}]| ≥ 2δ(G)− 3 as well as |NG[{v, v′}]| ≥ 2δ(G)− 3. Thus we obtain n(G) = 3δ(G)− 5 ≥ |NG[{u, u′}]|+ |NG[{v, v′}]| − |S| ≥ 4δ(G)− 6− |S| ≥ 4δ(G)− 6− (δ(G)− 1) = 3δ(G)− 5. This shows that G− S consists of exactly two components with vertex sets Wu and Wv such that u ∈ Wu and v ∈ Wv, |S| = δ(G)− 1 and |Wu| = |Wv| = δ(G)− 2. Subcase 2.1. Assume that δ(G) = 4. Then |S| = 3 and |Wu| = |Wv| = 2. This implies that each vertex of Wu∪Wv is adjacent to each vertex in S. Hence G contains a K2,4 as a subgraph, a contradiction. Subcase 2.2. Assume that δ(G) = 5. Then |S| = 4 and |Wu| = |Wv| = 3. Now we have the same situation as in Subcase 1.1.2. Hence G contains a K2,4 as a subgraph, a contradiction. Subcase 2.3. Assume that δ(G) = 6. Then |S| = 5 and |Wu| = |Wv| = 4. Now we have the same situation as in Subcase 1.1.3. Hence G contains a K2,4 as a subgraph, a contradiction. 36 ANDREAS HOLTKAMP AND LUTZ VOLKMANN Subcase 2.4. Assume that δ(G) ≥ 7. Then |Wu| ≥ 5. Let w1, w2, w3 ∈ Wu be three pairwise distinct vertices. Since G is K2,4-free, we observe that |NG[{w1, w2, w3}]| ≥ 3δ(G)− 9, and we arrive at the contradiction 3δ(G)−5 = n(G) ≥ |NG[{w1, w2, w3}]|+ |Wv| ≥ 4δ(G)−11 ≥ 3δ−4. Combining Theorem 7 with Observation 1, we obtain the next result immediately. Corollary 8 Let G be a connected K2,4-free graph with minimum degree δ ≥ 3. If n(G) ≤ 3δ(G)− 5, then G is maximally connected. The example in Figure 3 demonstrates that the bound given in Theorem 7 as well as in Corollary 8 is best possible, at least for δ = 4. Figure 3: K2,4-free graph with δ = 4 and n = 3δ − 4 = 8 vertices which is not maximally (local) connected.