Proper connection number and graph products *footnote **footnote *Supported by the National Science Foundation of China (No. 11161037) and the Science Found of Qinghai Province (No. 2014-ZJ-907).

Proper connection number and graph products ***Supported by the National Science Foundation of China (No. 11161037) and the Science Found of Qinghai Province (No. 2014-ZJ-907).

Yaping Mao,  Fengnan Yanling Corresponding author,  Zhao Wang,  Chengfu Ye
Department of Mathematics, Qinghai Normal
University, Xining, Qinghai 810008, China
Abstract

A path in an edge-colored graph is called a proper path if no two adjacent edges of are colored the same, and is proper connected if every two vertices of are connected by a proper path in . The proper connection number of a connected graph , denoted by , is the minimum number of colors that are needed to make proper connected. In this paper, we study the proper connection number on the lexicographical, strong, Cartesian, and direct product and present several upper bounds for these products of graphs.
Keywords: connectivity; vertex-coloring; proper path; proper connection number; direct product; lexicographic product; Cartesian product; strong product.
AMS subject classification 2010: 05C15, 05C40, 05C76.

1 Introduction

All graphs considered in this paper are simple, finite and undirected. We follow the terminology and notation of Bondy and Murty [3]. For a graph , we use , , , , , , , and to denote the vertex set, edge set, number of vertices, number of edges, connectivity, edge-connectivity, minimum degree and diameter of , respectively. The rainbow connections of a graph which are applied to measure the safety of a network are introduced by Chartrand, Johns, McKeon and Zhang [7]. Readers can see [7, 8, 9] for details. An edge-coloring of a graph is an assignment of colors to the edges of , one color to each edge of . Consider an edge-coloring (not necessarily proper) of a graph . We say that a path of is rainbow, if no two edges on the path have the same color. An edge-colored graph is rainbow connected if every two vertices are connected by a rainbow path. The minimum number of colors required to rainbow color a graph is called the rainbow connection number, denoted by . For more results on the rainbow connection, we refer to the survey paper [15] of Li, Shi and Sun and a new book [16] of Li and Sun.

If adjacent edges of are assigned different colors by , then is a proper (edge-)coloring. The minimum number of colors needed in a proper coloring of G is referred to as the chromatic index of G and denoted by . Recently, Andrews, Laforge, Lumduanhom and Zhang [1] introduce the concept of proper-path colorings. Let be an edge-colored graph, where adjacent edges may be colored the same. A path in is called a proper path if no two adjacent edges of are colored the same. An edge-coloring is a proper-path coloring of a connected graph if every pair of distinct vertices of is connected by a proper - path in . A graph with a proper-path coloring is said to be proper connected. If colors are used, then is referred to as a proper-path -coloring. The minimum number of colors needed to produce a proper-path coloring of is called the proper connection number of , denoted by .

Let be a nontrivial connected graph of order and size . Then the proper connection number of has the following bounds.

Furthermore, if and only if and if and only if is a star of order . For more details on the proper connection number, we refer to [1, 17, 21].

The standard products (Cartesian, direct, strong, and lexicographic) draw a constant attention of graph research community, see some recent papers [2, 27, 31, 34].

In this paper, we consider four standard products: the lexicographic, the strong, the Cartesian and the direct with respect to the proper connection number. Every of these four products will be treated in one of the forthcoming sections.

2 The Cartesian product

The Cartesian product of two graphs and , written as , is the graph with vertex set , in which two vertices and are adjacent if and only if and , or and . Clearly, the Cartesian product is commutative, that is, is isomorphic to .

Lemma 1

[13] Let and be two vertices of . Then

Theorem 1

Let and be connected graphs with and . Then

Moreover, the bound is sharp.

Proof. Without loss of generality, we assume . Suppose be a proper coloring of . Clearly, Since is connected, there is a path connecting and , say where . By the same reason, there is a path connecting and , say where . Now we give a coloring of using colors. To show that , we provide a proper-coloring of with colors as follows.

It suffices to check that there is a proper-path between any two vertices in . If or , then or , respectively, is a trivial one vertex path. We distinguish the following two cases to prove this theorem.

Case .

If is even, then we let be an arbitrary neighbor of . The path induced by the edges in

is proper -path in .

If is odd, then we let be an arbitrary neighbor of . The path induced by the edges in

is proper -path in .

Case .

If , then . Clearly, there is a proper-path connecting and . Now we consider . If is even, then we let be an arbitrary neighbor of . The path induced by the edges in

,

is proper -path in .

If is odd, then we let be an arbitrary neighbor of . The path induced by the edges in

is a proper -path in .  

To show the sharpness of the above bound, we consider the following example.

Example 1: Let and . Then by Theorem 1. From Lemma 1, we have and hence . Therefore, .

3 The strong product

The strong product of graphs and has the vertex set . Two vertices and are adjacent whenever and , or and , or and .

Lemma 2

[13] If is a nontrivial connected graph and is a connected spanning subgraph of , then .

The strong product is connected whenever both factors are and the vertex connectivity of the strong product was solved recently by Spacapan in [23].

By Lemma 2, we have . By Theorem 1, the following proposition is immediate.

Proposition 1

Let and be connected graphs. Then

Moreover, the bound is sharp.

Lemma 3

[13] Let and be two vertices of . Then

To show the sharpness of the upper bound in Proposition 1, we consider the following example.

Example 2: Let be a complete graph and . From Proposition 1, we have . By Lemma 3, and hence . Therefore, .

4 The lexicographical product

The lexicographic product of graphs and has the vertex set . Two vertices are adjacent if , or if and . The lexicographic product is not commutative and is connected whenever is connected.

In this section, let and be two connected graphs with and , respectively. Then . For , we use to denote the subgraph of induced by the vertex set . Similarly, for , we use to denote the subgraph of induced by the vertex set .

Theorem 2

Let and be connected graphs.

For , we have

If , then ;

If , then ;

If , then .

Moreover, the bound is sharp.

Proof. If , then we give a coloring of using colors. Suppose is a proper-coloring of . We color the edges the same as , and the edges . It suffices to check that there is a proper-path between any two vertices in . If , then there is a proper path in as desired. Now suppose . Since , there is an edge such that . The path induced by the edges in

is a proper-path connected and .

If , then by Lemma 2 and Theorem 1.

If , then we color as follows.

It suffices to check that there is a proper-path between any two vertices in . If , then there is a proper-path connecting and in , as desired. Suppose . If , then . There is a proper-path connecting and . We now assume . Since is connected, it follows that there is a proper-path connecting and in , say . Then the path induced by the edges in is a proper-path connecting and . Therefore, the above coloring is a proper-path coloring of , and hence .

If , then by Lemma 1 and Theorem 2. Since , . So

The same as .

If , then both and are complete. So .  

To show the sharpness of the upper bound in Theorem 2, we consider the following example.

Example 3: Let be a path of order and be a path of order . If , then , so by Theorem 2. Since , . So . If , , then , so by Theorem 2. Since , it follows that . So ; If , , then , then by Theorem 2. Since , we have . So ; If , then , and by Theorem 2.

Corollary 1

Let and be connected graphs, then .

5 The direct product

The direct product of graphs and has the vertex set . Two vertices and are adjacent if the projections on both coordinates are adjacent, i.e., and . It is clearly commutative and associativity also follows quickly. For more general properties we recommend [13]. The direct product is the most natural graph product in the sense of categories. But this also seems to be the reason that it is, in general, also the most elusive product of all standard products. For example, needs not to be connected even when both factors are. To gain connectedness of at least one factor must additionally be nonbipartite as shown by Weichsel [33]. Also, the distance formula

for the direct product is far more complicated as it is for other standard products. Here represents the length of a shortest even walk between and in , and the length of a shortest odd walk between and in . The formula was first shown in [25] and later in [19] in an equivalent version. There is no final solution for the connectivity of the direct product, only some partial results are known (see [4, 20]).

In this section we construct different upper bounds for the proper connection number of the direct product with respect to some invariants of the factors that are related to the rainbow vertex-connection number of the factors. A similar concept as for the distance formula is used and is due to the rainbow odd and even walks between vertices (and not only rainbow paths) and is thus, in a way, related with the formula. We say that is odd-even proper connected if there exists a proper colored odd path and a proper colored even path between every pair of (not necessarily different) vertices of . The odd-even proper connection number of a graph , , is the smallest number of colors needed for to be odd-even proper connected and it equals infinity if no such a coloring exists. A bipartite graph has either only even or only odd paths between two fixed vertices, thus there is no odd-even proper coloring of such a graph. On the other hand, let be a graph in which every vertex lies on some odd cycle. Then is finite since coloring every vertex with its own color produces an odd-even proper coloring of .

One can see that a odd cycle is an example where this coloring is optimal, and for a connected graph .

It is also easy to see that . For , and is odd, . For , and is even, .

Let be a graph. We split into two spanning subgraphs and , where the set consists of all edges of that lie on some odd cycle of , and the set . Clearly, and are not always connected. Let and be components of and , respectively, each one containing more than one vertex. Let

and

Note that is finite since it is defined on nontrivial components , .

Theorem 3

Let and be a nonbipartite connected graph. Then

Proof. Without loss of generality, . Denote by an optimal proper-coloring of components of . Let be an optimal odd-even proper-coloring of components of .

We give a proper-coloring of as follows. If projects on to , we set , and if projects on to , we set . where is the projection of on . By this way, we get a coloring of with colors and it remains to show that this is a rainbow coloring of .

Let and be arbitrary vertices from . Clearly, there is a proper path connecting and , say . By the same reason, there is a proper path connecting and , say . Observe that is a shortest proper -path in induced by and , and is a shortest proper -path in . If or , then or , respectively, is a trivial one vertex path.

We distinguish the following two cases to prove this theorem.

Case . and have the same parity.

If , then we let be an arbitrary neighbor of . Then the path induced by the edges in

is a proper-path in .

If , then we let be an arbitrary neighbor of . Then the path induced by the edges in

is a vertex-rainbow -path in .

If , and , then the path induced by the edges in

is a proper -path in whenever , and the path induced by the edges in

is a proper -path in whenever .

Case 2. and have different parity.

If there exists a -subpath of in , we replace this subpath by a rainbow -path of different parity in to obtain a proper path between and . If this is the case, then and have the same parity and we can use Case . We now assume that all the -subpaths of in , that is, all vertices of are in . To find a proper -path in , we find out a -walk in . Note that is contained in one component . Let be a vertex that is closest to any component of and let be closest to . Let be a shortest -path. From the definition of odd-even rainbow vertex-coloring, we know that there exists an odd vertex-rainbow -cycle in . Now we insert a closed walk that follows from into a path to obtain a -walk

of length . Note that and have different parity since is an odd number, and thus and have the same parity. If , then the path induced by the edges in

is a proper-coloring connected and .

If , then the path induced by the edges in

is a proper-coloring connected and .  

Corollary 2

Let and be connected graphs, where is nonbipartite and is bipartite. Then

A bipartite graph is said to have a property if admits of an automorphism such that if and only if . For more details, we refer to [23].

Lemma 4

[23] If and are bipartite graphs one of which has property , then the two components of are isomorphic.

Proposition 2

Let be a nonbipartite connected graph. Then

Proof. Let be an optimal odd-even proper-coloring of and let be an optimal proper-coloring of (for both cases it holds that no color appears in two different components). Observe that and . We provide a coloring of with colors as follows.

Recall that and are all the components of and , respectively. By the definition, is bipartite graph. From Lemma 3, can be decomposed into two subgraphs isomorphic to . Color both components of (which are isomorphic to ) optimally with colors for every . For this we use colors. Now, we assign new colors to the remaining vertices. For an edge of , it project on to an edge of receive color . For an edge of , it project on to an edge of receive color . For the introduced coloring colors are used and we need to show that is a proper-coloring of .

Set . Let and be arbitrary vertices in . Let be a proper -path under the proper-coloring of induced by and . We distinguish two cases to show this proposition.

Case . Let and have the same parity.

Without loss of generality we may assume that . Consequently if is an even number and otherwise. Thus

is a proper -path in .

Case . Let and have different parity.

Suppose first that has a nonempty intersection with some and let be the first and the last vertex of in . Then we can find a proper -path in with length of different parity as is the length of the -subpath of in . Replacing the -subpath of by this proper -path in we obtain a proper -path of the same parity as and we continue as in Case .

Suppose now that has an empty intersection with every , . Then is contained in for some , and and are in different components and of , respectively. Since is nonbipartite, there exists a vertex in some component of . Set . Take a proper path from to in , a proper odd path from to in , and a rainbow path from to in . This is a proper -path in since we have used different colors for , and .  

6 Applications

In this section, we demonstrate the usefulness of the proposed constructions by applying them to some instances of Cartesian and lexicographical product networks.

The following results will be used later.

Lemma 5

[13] Let and be two vertices of . Let denote the degree of vertex in . Then

6.1 Two-dimensional grid graph

A two-dimensional grid graph is an graph that is the graph Cartesian product of path graphs on and vertices. See Figure 1 for the case . For more details on grid graph, we refer to [5, 22]. The network is the graph lexicographical product of path graphs on and vertices. For more details on , we refer to [30]. See Figure 1 for the case .

Figure 1: Two-dimensional grid graph ; The network .

Proposition 3

For network , .

For network , when , when or or