# Some extremal results on the colorful monochromatic
vertex-connectivity of a graph^{1}^{1}1Supported by NSFC No.11371205, “973” program
No.2013CB834204, and PCSIRT.

###### Abstract

A path in a vertex-colored graph is called a vertex-monochromatic path if its internal vertices have the same color. A vertex-coloring of a graph is a monochromatic vertex-connection coloring (MVC-coloring for short), if there is a vertex-monochromatic path joining any two vertices in the graph. For a connected graph , the monochromatic vertex-connection number, denoted by , is defined to be the maximum number of colors used in an MVC-coloring of . These concepts of vertex-version are natural generalizations of the colorful monochromatic connectivity of edge-version, introduced by Caro and Yuster. In this paper, we mainly investigate the Erdős-Gallai-type problems for the monochromatic vertex-connection number and completely determine the exact value. Moreover, the Nordhaus-Gaddum-type inequality for is also given.

Keywords

: vertex-monochromatic path, -coloring, monochromatic vertex-connection number, Erdős-Gallai-type problem, Nordhaus-Gaddum-type problem

AMS subject classification 2010

: 05C15, 05C35, 05C38, 05C40.

## 1 Introduction

All graphs considered in this paper are simple, finite, undirected and connected. We follow the terminology and notation of Bondy and Murty [3]. For a graph , we use , , , , , , to denote its vertex set, edge set, the number of vertices, the number of edges, maximum degree, minimum degree and the degree of vertex , respectively. For , let be the number of vertices in , and the subgraph of induced by . We use to denote the distance between two vertices and in , and to denote the maximum distance of any two vertices in . A -path is a path connecting and . A -geodesic is a -path of length . We write if is adjacent to , and if is not adjacent to .

A path in an edge-colored graph is a monochromatic path if all the edges on the path are colored the same. An edge-coloring of a graph is a monochromatical connection coloring (MC-coloring, for short) if there is a monochromatic path joining any two vertices in the graph. For a connected graph , the monochromatical connection number, denoted by , is defined to be the maximum number of colors used in an MC-coloring of . An extremal MC-coloring is an MC-coloring that uses colors. These concepts were introduced by Caro and Yuster in [5], where they obtained some nontrivial lower and upper bounds for . In [4], we studied two kinds of Erdős-Gallai-type problems for and completely solved them.

As a natural idea, we introduce the vertex-version of these concepts in the following. A path in a vertex-colored graph is a vertex-monochromatic path if its internal vertices have the same color. An vertex-coloring of a graph is a monochromatical vertex-connection coloring (MVC-coloring, for short), if there is a vertex-monochromatic path joining any two vertices in the graph. For a connected graph , the monochromatical vertex-connection number, denoted by , is defined to be the maximum number of colors used in an MVC-coloring of . An extremal MVC-coloring is an MVC-coloring that uses colors.

It is worth mentioning that the question for determining the monochromatic vertex-connection number is a natural opposite counterpart of the recently well-studied problem of vertex-rainbow connection number [16, 18, 9], where in the latter we seek to find the minimum number of colors needed in a vertex-coloring so that there is a vertex-rainbow path joining any two vertices.

An important property of an extremal MVC-coloring is that the vertices with each color form a connected subgraph. Indeed, if the subgraph formed by the vertices with a same color is disconnected, then a new color can be assigned to all the vertices of some component while still maintaining an MVC-coloring. For a color , the color subgraph is the connected subgraph of induced by the vertices with color . The color is nontrivial if has at least two vertices. Otherwise, is trivial. A nontrivial color subgraph with vertices is said to waste colors.

In this paper, we mainly investigate the Erdős-Gallai-type and Nordhaus-Gaddum-type results for colorful monochromatic vertex-connectivity of a graph.

The Erdős-Gallai-type problem is a kind of extremal problems to determine the maximum or minimum value of a graph parameter with some given properties. The interested readers can see the monograph written by Bollobs [2], which has a collection of such extremal problems in graph theory.

A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of the values of a parameter for a graph and its complement. The name “Nordhaus-Gaddum-type” is given because Nordhaus and Gaddum [19] first established the type of inequalities for the chromatic number of graphs in 1956. They proved that if and are complementary graphs on vertices whose chromatic numbers are and , respectively, then . Since then, many analogous inequalities of other graph parameters have been considered, such as diameter [13], domination number [14], rainbow connection number [8], and so on [7, 17]. For a good survey we refer to [1].

The rest of this paper is organized as follows. First, we prove some upper and lower bounds for in terms of the minimum degree and the diameter. Then we investigate the Erdős-Gallai-type problem and completely determine the exact value. Finally, the Nordhaus-Gaddum-type inequality for is given.

## 2 Upper and lower bounds for

For a connected graph , we take a spanning tree of . Color all the non-leaves in with one color, and each leave in with a distinct fresh color. Clearly, this is an MVC-coloring of with colors, where is the number of leaves in . Thus we get the following proposition.

###### Proposition 2.1.

Let be a connected graph with a spanning tree . Then .

In order to obtain a good lower bound for , we need to find a spanning tree with as many leaves as possible. By the known results about spanning trees with many leaves in [6, 11, 15], we have

###### Proposition 2.2.

Let be a connected graph on vertices with minimum degree .

If , then .

If , then .

If , then .

If , then .

We proceed with a lower bound for .

###### Proposition 2.3.

Let be a connected graph with vertices and diameter .

if and if only ;

If , then , and the bound is sharp.

###### Proof.

(1) holds obviously. For (2), the vertex-monochromatic path between the two vertices at distance wastes at least colors. Then . For the sharpness, we can take the graph obtained from a copy of by attaching a path of length at a vertex in . Clearly, . Give and the internal vertices on one color, and each other vertex in a distinct fresh color. It is easy to check that this vertex-coloring is an MVC-coloring of using colors, which implies . Thus . ∎

## 3 Erdős-Gallai-type results for

The following problems are called Erdős-Gallai-type problems.

Problem I: Given two positive integers , with , compute the minimum integer such that if a connected graph satisfies and , then .

Problem II: Given two positive integers , with , compute the maximum integer such that if a connected graph satisfies and , then .

Note that does not exist for , and , since for a star on vertices, we have . For this reason, the rest of the section is devoted to studying Problem I.

First, we state some lemmas, which are used to determine the value of .

###### Lemma 3.1.

[10] Let be a connected graph with and . Then has a spanning tree with at least leaves, and this is best possible.

###### Lemma 3.2.

[12] The maximum diameter among all connected graphs with vertices and edges is , where , , if for some , and otherwise.

###### Lemma 3.3.

Let be a cycle of order . Then

###### Proof.

For , we know , and thus . For , it is easy to check that . For , by Proposition 2.1, it suffices to prove that . By contradiction, we assume that . Let be an extremal -coloring of , and the color of vertex . Let . Since is monochromatically vertex-connected, for the pair of antipodal vertices , there exists a vertex-monochromatic path of length at least connecting them. Without loss of generality, suppose . Then , and we can find three vertices with three different colors but color . Then there exist no vertex-monochromatic paths connecting and , a contradiction. ∎

###### Lemma 3.4.

Let be the graph obtained from a complete graph on by replacing the edge with a path . Then for .

###### Proof.

Suppose that is an extremal -coloring of , and is the color of the vertex . Let , and . Denote by the set of all pairs of vertices in except , such that all the vertex-monochromatic -paths contain some vertex in . We call a path with color , if all the internal vertices on the path are colored by .

Case 1: .

Then for each pair of vertices in except , all the vertex-monochromatic -paths are contained in . Let be any vertex in . For each , the shortest vertex-monochromatic -paths must be or , which is contained in the cycle . For , is a vertex-monochromatic -path contained in . Thus induces an -coloring of a cycle for each .

Case 2: .

Then for with , the shortest vertex-monochromatic -paths must be , where is some vertex in with or .

Suppose first . Then we can find a vertex in such that . Such vertex must exist; otherwise there are no vertex-monochromatic paths connecting the pairs of vertices in . For each , is a vertex-monochromatic -path. With similar arguments as in Case 1, we get that induces an -coloring of the cycle .

Now suppose , say . Then for , exactly one of must be or ; otherwise, the vertex-monochromatic -paths contain both and as internal vertices, but , a contradiction. For , let be the set of pairs of vertices in containing . If one of is empty, say and , then we assume that and is a vertex-monochromatic -path, where . Obviously, is with color blue. For each , is a vertex-monochromatic -path. With similar arguments as in Case 1, we get that induces an -coloring of the cycle .

Now consider the case and . Assume that and . Let (resp. ) be a vertex-monochromatic path connecting (resp. ), where . Obviously, is with color blue, while is with color red. We claim that . Otherwise, both and contain as an internal vertex, but and are with different colors, a contradiction. Now we recolor all the vertices in colored by blue except by red, and get a new vertex-coloring . Next we will show that is still an extremal -coloring. It suffices to consider the pairs of vertices which only have vertex-monochromatic paths with color blue in . Let be such a pair, and a shortest vertex-monochromatic -path with color blue in . If does not contain as an internal vertex, then is a vertex-monochromatic -path with color red in . Otherwise, must have the form (). Now take the path , which is a vertex-monochromatic -path with color red in . Thus is an extremal -coloring of , in which the vertices receive the same color. This is the case we have discussed.

Therefore we come to the conclusion that there exists an extremal -coloring of , which induces an -coloring of a cycle for some . Since the cycle has length , we have by Lemma 3.3. So . ∎

###### Lemma 3.5.

Let be a connected graph with vertices and edges. Then , and this bound is sharp.

###### Proof.

If , then has a spanning tree with at least leaves. Hence . We are done. Now we assume . It follows from Lemma 3.2 that . If , then by Proposition 2.3. We are done. Now we assume . If contains only one pair of vertices at distance 3, then give the two internal vertices of a -geodesic one color, and each other vertex a different fresh color. Clearly, it is an -coloring of using colors. Thus . We are done. Now suppose that contains at least two pairs of vertices at distance 3. If there exists two pairs of vertices at distance 3 such that , then are not adjacent and have no common neighbors, since . So we have for . Thus . On the other hand, , a contradiction. Now suppose that for any two pairs of vertices at distance 3, . We distinguish the following cases.

Case 1: All the pairs of vertices at distance 3 have a common vertex, say .

Since , it follows from Lemma 3.1 that has a spanning tree with at least leaves. Hence . By contradiction, we assume that . Let be an extremal -coloring of , and be the color of vertex . Thus wastes two colors. This can be classified into the following two subcases:

Subcase 1.1: There are two nontrivial colors and , and the color subgraph (resp. ) consists of two adjacent vertices (resp. ).

Then for each pair of vertices at distance 3, must be connected by a vertex-monochromatic path with color or . Let be the set of vertices with such that can be connected by a vertex-monochromatic path with color , say (this implies ). Let be the set of vertices with such that can only be connected by a vertex-monochromatic path with color , say (this implies ). See Fig 1. Clearly, and ; otherwise we can get an -coloring using more colors. Moreover, is a partition of all the vertices at distance 3 from .

Let . For , if , then is not adjacent to any vertex in , since the distance between them is 3. If , then can not be adjacent to every vertex in ; otherwise we can give one color, and each other vertex a distinct fresh color, which is an -coloring using colors. Thus is not adjacent to at least two vertices in . For , since , is not adjacent to . By the definition of , is not adjacent to any vertex in . Furthermore, can not be adjacent to all the vertices in ; otherwise we can give a fresh color, and get an -coloring using colors. As we have noted, is not adjacent to , . From the above, we have , a contradiction.

Subcase 1.2: There is exactly one nontrivial color , and the color subgraph consists of three vertices .

For some pair of vertices at distance 3, they are connected by a vertex-monochromatic path with color . Without loss of generality, we assume (this implies ). For , there must exist a pair of vertices at distance 3 such that all the vertex-monochromatic paths connecting them contain . If , then , since and , a contradiction. If , then is also a vertex-monochromatic -path not containing , a contradiction. If , then , a contradiction. Thus must be the the form (this implies ). Let be the set of vertices with such that are connected by a vertex-monochromatic path . Let be the set of vertices with such that can only be connected by a vertex-monochromatic path . See Fig 1. Clearly, and . Moreover, is a partition of all the vertices at distance 3 from .

Let . With similar arguments as in Subcase 1.1, we have (1) For , is not adjacent to at least two vertices in . (2) For , is not adjacent to . (3) is not adjacent to any vertex in . (4) is not adjacent to all the vertices in . (5) is not adjacent to , . From the above, we have , a contradiction.

Therefore, in Case 1 we have .

Case 2: There exist three pairs of vertices with , such that .

Since any two such pairs have a common vertex, without loss of generality, we may assume . Now the three pairs can be written as . As two vertices in each pair are at distance 3, , , , and each vertex in is adjacent to at most one vertex in . Thus . Then we have . On the other hand, , a contradiction.

Now we show the sharpness of the bound. Let be the graph obtained from a complete graph on . by adding a path to it. It is easily checked that and . By Proposition 2.3, we know . Hence . ∎

###### Theorem 3.6.

Let be a connected graph with vertices and edges. If for , then , and this bound is sharp except for . For the latter two cases, , and this bound is sharp.

###### Proof.

Let . Then .

Case 1: .

If , then it follows from Lemma 3.2 that the diameter of is at most . If , then the diameter of is at most . By Proposition 2.3, we have .

Case 2: .

By Lemma 3.1, we know that contains a spanning tree with at least leaves. Then .

Next we will show the sharpness of the bound. If , then we can take the extremal graph as follows: First take a complete graph with vertex set , and then add a path to it, and finally add the remaining edges (at most ) between and randomly. It is easily checked that . By Proposition 2.3, we have . Hence . If and , then we can take the extremal graph as in Lemma 3.4. It is easily checked that , and . By Lemma 3.4, we have . Hence . If and , then , i.e. . Thus .

If and , then . Now by Lemma 3.5, we have , and this bound is sharp.

If and , then it follows from Lemma 3.2 that the maximum diameter is . Hence . ∎

###### Corollary 3.7.

Given two integers with ,

###### Proof.

Since for any connected graph , we know . For , if , then it follows from Theorem 3.6 that . Hence . For , by Theorem 3.6, there exists a graph with vertices and edges such that . Hence . So we get for . For , if , then it follows from Theorem 3.6 that . Hence . For , by Theorem 3.6, there exists a graph with vertices and edges such that . Hence . So we get for . ∎

## 4 Nordhaus-Gaddum-type theorem for

A double star is a tree with diameter 3. The centers of a double star are the two nonleaves in it.

###### Lemma 4.1.

[20] Let be a connected graph with connected complement . Then

if , then ,

if , then has a spanning subgraph which is a double star.

As we all know, a connected graph on vertices has at least edges. If both and are connected, then , and so . In the sequel, we always assume that has at least vertices, and both and are connected. Clearly, for , both and are a path on four vertices. Thus , and .

###### Theorem 4.2.

If is a graph on vertices, then , and the bounds are sharp.

###### Proof.

For any graph , we have a trivial upper bound . So . Now take the graph in Fig 2.

It is easily checked that . By Proposition 2.3, we have , which implies the sharpness of the bound.

For the lower bound, if , then by Lemma 4.1, we have . Hence . Now we can suppose and . If and , then similarly we have . If , then by Lemma 4.1, (resp. ) contains a double star (resp. ) as a spanning subgraph. And , since we can give the two centers in one color, and each other vertex a distinct fresh color, which induces an -coloring using colors. Thus for . Now we construct a graph that reaches the lower bound. Just take . Since , it follows from Lemma 4.1 that . Then . The proof is complete. ∎

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