The minimal size of a graph with given generalized 3-edge-connectivity1footnote 11footnote 1Supported by NSFC No.11071130

The minimal size of a graph with given generalized -edge-connectivity111Supported by NSFC No.11071130

Xueliang Li, Yaping Mao
Center for Combinatorics and LPMC-TJKLC
Nankai University, Tianjin 300071, China
lxl@nankai.edu.cn; maoyaping@ymail.com.
Abstract

For and , is the maximum number of edge-disjoint trees connecting in . For an integer with , the generalized -edge-connectivity of is then defined as . It is also clear that when , is nothing new but the standard edge-connectivity of . In this paper, graphs of order such that is characterized. Furthermore, we determine the minimal number of edges of a graph of order with and give a sharp lower bound for .

Keywords

: edge-connectivity, Steiner tree, edge-disjoint trees, generalized edge-connectivity.
AMS subject classification 2010: 05C40, 05C05, 05C75.

1 Introduction

All graphs considered in this paper are undirected, finite and simple. We refer to the book [1] for graph theoretical notation and terminology not described here. For a graph , let and denote the set of vertices and the set of edges of , respectively. As usual, the union of two graphs and is the graph, denoted by , with vertex set and edge set . Let be the disjoint union of copies of a graph . We denote by the set of edges of with one end in and the other end in . If , we simply write for .

The generalized connectivity of a graph , introduced by Chartrand et al. in [2], is a natural and nice generalization of the concept of (vertex-)connectivity. For a graph and a set of at least two vertices, an -Steiner tree or a Steiner tree connecting (or simply, an -tree) is a such subgraph of that is a tree with . Two Steiner trees and connecting are said to be internally disjoint if and . For and , the generalized local connectivity is the maximum number of internally disjoint trees connecting in . Note that when a Steiner tree connecting is just a path connecting the two vertices of . For an integer with , the generalized -connectivity of is defined as . Clearly, when , is nothing new but the connectivity of , that is, , which is the reason why one addresses as the generalized connectivity of . By convention, for a connected graph with less than vertices, we set . Set when is disconnected. Results on the generalized connectivity can be found in [3, 4, 6, 7, 8, 10, 13].

As a natural counterpart of the generalized connectivity, we introduced the concept of generalized edge-connectivity in [11]. For and , the generalized local edge-connectivity is the maximum number of edge-disjoint trees connecting in . For an integer with , the generalized -edge-connectivity of is then defined as . It is also clear that when , is nothing new but the standard edge-connectivity of , that is, , which is the reason why we address as the generalized edge-connectivity of . Also set when is disconnected.

In addition to being natural combinatorial measures, the generalized connectivity and generalized edge-connectivity can be motivated by their interesting interpretation in practice. For example, suppose that represents a network. If one considers to connect a pair of vertices of , then a path is used to connect them. However, if one wants to connect a set of vertices of with , then a tree has to be used to connect them. This kind of tree with minimum order for connecting a set of vertices is usually called a Steiner tree, and popularly used in the physical design of VLSI, see [14]. Usually, one wants to consider how tough a network can be, for connecting a set of vertices. Then, the number of totally independent ways to connect them is a measure for this purpose. The generalized -connectivity and generalized -edge-connectivity can serve for measuring the capability of a network to connect any vertices in .

The following two observations are easily seen.

Observation 1.

If is a connected graph, then .

Observation 2.

If is a spanning subgraph of , then and .

In [11], we obtained some results on the generalized edge-connectivity. The following results are restated, which will be used later.

Lemma 1.

[11] For every two integers and with ,

Lemma 2.

[11] For any connected graph , . Moreover, the upper bound is sharp.

Lemma 3.

[11] Let be two integers with . For a connected graph of order , . Moreover, the upper and lower bounds are sharp.

In [11], we characterized graphs with large generalized -connectivity and obtained the following result.

Lemma 4.

[11] Let be two integers with . For a connected graph of order , if and only if for even; for odd, where is an edge set such that .

Like [5], here we will consider the generalized -edge-connectivity. From Lemma 3, . In Section , graphs of order such that is characterized.

Let be the minimal number of edges of a graph of order with . From Lemma 4, we know that for even; for odd. It is not easy to determine exact value of the parameter . So we put our attention to on the case . The exact value of for are obtained in Section . We also give a sharp lower bounds of for general .

2 Graphs with

After the preparation of the above section, we start to give our main result. From Lemma 3, we know that for a connected graph of order . Graphs with has been shown in Lemma 4. But, it is not easy to characterize graphs with for general . So we focus on the case that and characterizing the graphs with in this section.

For the generalized -connectivity, we got the following result in [5].

Theorem 1.

[5] Let be a connected graph of order . if and only if or or or .

But, for the edge case we will show that the statement is different. Before giving our main result, we need some preparations.

Choose . Then let be a maximum set of edge-disjoint trees connecting in . Let be the set of trees in whose edges belong to , and let be the set of trees containing at least one edge of . Thus .

In [11], we obtained the following useful lemma.

Lemma 5.

[11] Let , and be a tree connecting . If , then uses edges of ; If , then uses at least edges of .

By Lemma 6, we can derived the following result.

Lemma 6.

Let be a connected graph of order , and be a positive integer. If we can find a set with satisfying one of the following conditions, then .

and ;

and ;

and ;

and .

Proof.

We only show that and hold, and can be proved similarly.

Since , we have . Since , we have . Therefore, , and so there exists at most one tree belonging to in . If there exists one tree belonging to , namely , then the other trees connecting must belong to . From Lemma 6, each tree belonging to uses at least edges in . So the remaining at most edges of can form at most trees. Thus , which results in since is an integer. Suppose that all trees connecting belong to . Then , which implies that .

Since , we have . Since , we have . Since , there exists no tree belonging to . So each tree connecting must belong to . From Lemma 6, , which implies that since is an integer. ∎

Lemma 7.

Let be a connected graph with minimum degree . If there are two adjacent vertices of degree , then .

Proof.

It is clear that and by Lemma 2. So .

Suppose that there are two adjacent vertices and of degree and . Besides and , we choose a vertex in to get a -set containing . Suppose are pairwise edge-disjoint trees connecting . Since is simple graph, obviously the edges incident must be contained in , respectively, and so are the edges incident . Without loss of generality, we may assume that the edge is contained in . But, since is a tree connecting , it must contain another edge incident with or , a contradiction. Thus . ∎

A subset of is called a matching of if the edges of satisfy that no two of them are adjacent in . A matching saturates a vertex , or is said to be -saturated, if some edge of is incident with ; otherwise, is -unsaturated. is a maximum matching if has no matching with .

Theorem 2.

Let be a connected graph of order . Then if and only if or or or .

Proof.

Sufficiency: Assume that . From Lemma 4, for a connected graph , if and only if . Since , it follows that . We claim that . Assume, to the contrary, that . Then , a contradiction. Since , it follows that each component of is a path or a cycle (note that a isolated vertex in is a trivial path). We will show that the following claims hold.

Claim 1.   has at most one component of order larger than .

Suppose, to the contrary, that has two components of order larger than , denoted by and (see Figure 1 ).

Let and such that and is adjacent to in . Thus . The same is true for , that is, . Pick . This implies that . Since all other components of are paths or cycles, . So and hence . Since , by Lemma 7 it follows that , a contradiction.

Claim 2.  If is the component of of order larger than , then is a -path.

Assume, to the contrary, that is a path or a cycle of order larger than , or a cycle of order .

First, we consider the former. We can pick a in . Let , and (see Figure 1 ). Since , there exists no tree of type connecting . From Lemma 5, each tree of type uses at least edges. Since , we have and hence since is an integer. This contradicts to .

Figure 1: Graphs for Claims and .

Now we consider the latter. Let be a cycle, and (see Figure 1 ). Since , there exists no tree of type . Since each tree of type uses at least edges and , we have and , which also contradicts to .

From the above two claims, we know that if has a component , then it is the only component of order larger than and the other components must be independent edges. Let be the number of such independent edges. can have as many as such independent edges, which implies that . From Lemma 4, . Thus .

By the similar analysis, we conclude that or or or .

Necessity: We will show that if is a graph with the conditions of this theorem. We have the following cases to consider.

Case 1.   or .

We only need to show that for . If for , then for . It suffices to check that for .

Let and be a -subset of , and . It is clear that is a maximum matching of . Then has at most one -unsaturated vertex.

Figure 2: Graphs for Case .

If , then there exist pairwise edge-disjoint trees connecting since each vertex in is adjacent to every vertex in . Suppose .

If , then one element of belongs to , denoted by . Since , we can assume that , . When is -unsaturated, the trees together with form pairwise edge-disjoint trees connecting , where . When is -saturated, we let be the adjacent vertex of under . Then the trees together with and form pairwise edge-disjoint trees connecting (see Figure 2 ), where .

If , then two elements of belong to , denoted by and . Without loss of generality, let . When and are adjacent under , the trees together with and form pairwise edge-disjoint trees connecting (see Figure 2 ), where . When and are nonadjacent under , we consider whether and are -saturated. If one of is -unsaturated, without loss of generality, we assume that is -unsaturated. Since has at most one -unsaturated vertex, is -saturated. Let be the adjacent vertex of under . Then the trees together with and and form pairwise edge-disjoint trees connecting (see Figure 2 ), where . If both and are -saturated, we let be the adjacent vertex of under , respectively. Then the trees together with , , and form pairwise edge-disjoint trees connecting (see Figure 2 ), where .

Otherwise, . When one of is -unsaturated, without loss of generality, we assume that is -unsaturated. Since has at most one -unsaturated vertex, both and are -saturated. Let be the adjacent vertex of under , respectively. We pick a vertex of . When are all -saturated, we let be the adjacent vertex of under , respectively. Then the trees together with and and and form pairwise edge-disjoint trees connecting (see Figure 2 ), where .

From the above discussion, we get that for , which implies . So .

Case 2.   or .

We only need to show that for and . If for , then for . So we only need to consider the former. Let , be a -subset of , and . Clearly, is a maximum matching of . It is easy to see that has at most one -unsaturated vertex. For any , we will show that there exist edge-disjoint trees connecting in .

If , then there exist pairwise edge-disjoint trees connecting since each vertex in is adjacent to every vertex in . Since and , we only need to consider and . These trees together with for , or for form pairwise edge-disjoint trees connecting . Suppose .

Figure 3: Graphs for in Case 2.

If , then one element of belongs to , denoted by . Since and , we only need to consider or or . When is -unsaturated, the trees together with , for , or for , or for form pairwise edge-disjoint trees connecting , where . When is -unsaturated, we let be the adjacent vertex of under . For , the trees together with , and form pairwise edge-disjoint trees connecting (see Figure 3 ), where . One can check that the same is true for and (see Figure 3 and ).

If , then two elements of belong to , denoted by and . We only need to consider or . When and are adjacent under , the trees together with , and form pairwise edge-disjoint trees connecting for (see Figure 3 ), where . The same is true for (see Figure 3 ). When and are nonadjacent under , we consider whether and are -saturated. If one of is -unsaturated, without loss of generality, we assume that is -unsaturated. Since has at most one -unsaturated vertex, is -saturated. Let be the adjacent vertex of under . For