A note on the saturation number of the family of k-connected graphs

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1 A note on the saturation number of the family of k-connected graphs Paul S. Wenger January 8, 014 Abstract Given a family of graphs F, a graph G is F-saturated if no member of F is a subgraph of G, but for all e E(G), some member of F is a subgraph of G + e. The saturation number of F, denoted sat(n, F), is the minimum number of edges in an n-vertex F-saturated graph. In this note we determine the saturation number for the family of k-connected graphs. Keywords: 05C35; saturation; k-connected; subgraph Given a family of graphs F, a graph G is F-saturated if no member of F is a subgraph of G, but for all e E(G), some member of F is a subgraph of G + e. The maximum number of edges in an n-vertex F-saturated graph is the traditional extremal number introduced by Turán. The saturation number of F, denoted sat(n, F), is the minimum number of edges in an n-vertex F-saturated graph. Saturation numbers were first studied by Erdős, Hajnal, and Moon [3], who proved that sat(n, K k ) = (k )n ( ) k 1 for n k. Furthermore, they proved that equality holds only for the graph G n,k consisting of a complete graph Q with k vertices plus n k + independent vertices adjacent to every vertex of Q. For a thorough account of the results known about saturation numbers, the reader should consult the excellent survey of Faudree, Faudree, and Schmitt [5]. Let F k denote the family of k-connected graphs. In [9], Mader proved that every n- vertex graph with at least kn edges contains a k-connected graph, thus establishing an upper bound on the extremal number of F k. This bound was later improved by Yuster [1] to 193 kn. These results have become quite useful, notably pertaining to graph minors and 10 subdivisions (e.g. [1, 7, 8]) and various partition problems (e.g. [, 4, 6]). In this note we consider the saturation variant of this question and determine sat(n, F k ). Observe that the School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY; pswsma@rit.edu. 1

2 condition that a graph G is F k -saturated does not mean that G is a maximal graph that is not k-connected. Rather, the addition of an edge to G makes a k-connected subgraph. Throughout we use the terminology and notation of [11]. A k-tree is any graph that is either K k or is obtained by joining a new vertex to a k-clique in a k-tree. Theorem 1. For n k + 1, sat(n, F k ) = (k 1)n. Furthermore, every (k 1)-tree with n vertices is F k -saturated and has this number of edges. The unique smallest K k+1 -saturated graph with n vertices is the Erdős-Hajnal-Moon graph K k 1 K n k+1 ; it is in fact a (k 1)-tree. We prove that all (k 1)-trees are F k - saturated in order to show that the family of minimum F k -saturated graphs is richer than the family of minimum K k+1 -saturated graphs. However, the family of (k 1)-trees is not the complete family of minimum F k -saturated graphs. For example, K k+ (P 3 K ) is F k -saturated but is not a (k 1)-tree. To prove that (k 1)-trees are F k -saturated, we use the following characterization of k-trees and corollary due to Rose [10]. Given two vertices x and y in a graph G, an x, y- separator is a set of vertices S such that G S has no x, y-path. Theorem (Theorem 1.1 in [10]). A graph G is a k-tree if and only if (i) G is connected, (ii) G has a k-clique but no (k + )-clique, and (iii) for all x, y V (G), every minimal x, y-separator is a k-clique. A set of vertices S in a graph G is a separating set if G S has more than one component. The connectivity of G, denoted κ(g), is the minimum size of a separating set when G is not complete and n 1 when G = K n. Given a separating set S in a graph G, an S-lobe is a subgraph of G induced by the union S and the vertex set of a component of G S. Corollary 3 (Corollary.3 in [10]). Let G be a k-tree. If S is a clique that is a separating set of G, then each S-lobe of G is a k-tree. Proof of Theorem 1. Lower Bound: First we show that sat(n, F k ) (k 1)n ( k ). We proceed by induction on n. Observe first that the bound holds trivially for n = k + 1, since the only k-connected graph with at most k + 1 vertices is K k+1. Let G be a F k -saturated graph with V (G) k +.

3 Let S be a smallest separating set in G. Let X and Y be two components in G S, with x V (X) and y V (Y ). Since G is F k -saturated, there is a k-connected subgraph in G + xy, and this subgraph must contain the edge xy. By Menger s Theorem, there are k 1 internally disjoint x, y-paths in G that pass through S. Thus S k 1. Since G is not k-connected, we conclude that κ(g) = k 1. Let S be a smallest separating set in G; hence S = k 1. For any xy E(G) such that x and y lie in the same S-lobe of G (they may both lie in S), a k-connected subgraph H of G + xy must lie in a single S-lobe of G. Otherwise, H contains two vertices separated by S. We now consider two subgraphs of G. Let Y be a component of G S, and let G 1 be the S-lobe containing Y. Let G = G V (Y ). Let m be the number of edges induced by S. If S contains nonadjacent vertices x and y, then G 1 + xy or G + xy contains a k-connected subgraph. If G 1 + xy does not contain a k-connected subgraph, then modify G 1 by adding the edge xy. Similarly, if G + xy does not contain a k-connected subgraph, then modify G by adding the edge xy. Repeat this process for each pair of nonadjacent vertices in S. Thus, with the addition of at most ( ) k 1 m edges, we obtain two Fk -saturated graphs G 1 and G. Let V (G 1 ) = a + k 1 and let V (G ) = b + k 1; thus V (G) = a + b + k 1. By the induction hypothesis we conclude that E(G 1) (k 1)(a + k 1) ( k ) and E(G ) (k 1)(b + k 1) ( k ). Since each of the m edges induced by S lies in both G 1 and G, we obtain the following count on the number of edges in G: 1 E(G) E(G 1) + E(G ) (k 1)(a + k 1) + (k 1)(b + k 1) = (k 1) V (G). 1 Upper Bound: To prove that sat(n, F k ) (k 1)n ( k ) we show that all (k 1)-trees are F k -saturated. We use induction on n. The result holds trivially for n = k + 1 since the only k-connected graph with at most k + 1 vertices is K k+1. Let G be an n-vertex (k 1)-tree with n k +. It is easy to show by induction that G has at least two vertices of degree k 1, and the vertices of degree k 1 form an independent set. Let x be a vertex in G with d(x) = k 1; it follows that G x is also a (k 1)-tree. By the induction hypothesis, G x is F k -saturated, so it remains to show that adding an edge to G that is incident to x yields a k-connected subgraph. 3

4 Let y be a nonneighbor of x in G, and consider the addition of the edge xy. If G has a vertex z such that z / {x, y} and G z is a (k 1)-tree, then by induction G + xy contains a k-connected subgraph. Thus we may assume that G has exactly two vertices of degree k 1, namely x and y. Let S be a set of k 1 vertices in G + xy. If S is not a clique, then by Theorem, G S is connected, and therefore S is not a separating set of G + xy. If S is a clique that separates G, then by Corollary 3 each S-lobe of G is a (k 1)-tree. Thus each lobe contains a vertex of degree k 1 (in G) that is not in S. Consequently there are only two S-lobes in G, one containing x and the other containing y. Thus S is not a separating set in G + xy, and we conclude that G + xy is k-connected. References [1] T. Böhme, K-I. Kawarabayashi, J. Maharry, and B. Mohar, Linear connectivity forces large bipartite minors, J. Combin. Theory Ser. B 99 (009), no. 3, [] T. Böhme and A. Kostochka, Many disjoint dense subgraphs versus large k-connected subgraphs in large graphs with given edge density, Discrete Math. 309 (009), no. 4, [3] P. Erdős, A. Hajnal and J.W. Moon, A problem in graph theory, Amer. Math. Monthly 71 (1964), [4] M. Ferrara, C. Magnant, P. Wenger, Conditions for families of disjoint k-connected subgraphs in a graph, Discrete Math. 313 (013), no. 6, [5] J. Faudree, R. Faudree, and J. Schmitt, A survey of minimum saturated graphs, Electron. J. Combin. 18 (011), Dynamic Survey 19, 36 pp. (electronic). [6] A. Gyárfás, M. Ruszinkó, G. Sárközy, and E. Szemerédi, An improved bound for the monochromatic cycle partition number. J. Combin. Theory Ser. B 96 (006), no. 6, [7] D. Kühn, and D. Osthus, Forcing unbalanced complete bipartite minors, European J. Combin. 6 (005), no. 1, [8] D. Kühn, and D. Osthus, Extremal connectivity for topological cliques in bipartite graphs, J. Combin. Theory Ser. B 96 (006), no. 1,

5 [9] W. Mader, Existenz n-fach zusammenhängender Teilgraphen in Graphen genügend grosser Kantendichte, Abh. Math. Sem. Univ. Hamburg 37 (197), [10] D.J. Rose, On simple characterizations of k-trees, Discrete Math., 7 (1947), [11] D.B. West, Introduction to Graph Theory, second ed., Prentice Hall, Upper Saddle River, NJ, 001. [1] R. Yuster, A note on graphs with k-connected subgraphs, Ars Combin. 67 (003),