(g,f)-Chromatic spanning trees and forests

09/27/2018 ∙ by Kazuhiro Suzuki, et al. ∙ 0

A heterochromatic (or rainbow) graph is an edge-colored graph whose edges have distinct colors, that is, where each color appears at most once. In this paper, I propose a (g,f)-chromatic graph as an edge-colored graph where each color c appears at least g(c) times and at most f(c) times. I also present a necessary and sufficient condition for edge-colored graphs (not necessary to be proper) to have a (g,f)-chromatic spanning tree. Using this criterion, I show that an edge-colored complete graph G has a spanning tree with a color probability distribution `similar' to that of G. Moreover, I conjecture that an edge-colored complete graph G of order 2n (n > 3) can be partitioned into n edge-disjoint spanning trees such that each has a color probability distribution `similar' to that of G.

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1 Introduction

We consider finite undirected graphs without loops or multiple edges. For a graph , we denote by and its vertex and edge sets, respectively. An edge-coloring of a graph is a mapping , where is a set of colors. Then, the triple is called an edge-colored graph. We often abbreviate an edge-colored graph as . Note that an edge colored graph is not necessary to be proper, where distinct red edges may have a common end vertex.

1.1 Heterochromatic (or rainbow) spanning trees

An edge-colored graph is said to be heterochromatic222A heterochromatic graph is also said to be rainbow, multicolored, totally multicolored, polychromatic, or colorful, and so on. if no two edges of have the same color, that is, for any two distinct edges and of . As far as I know, there are three topics about heterochromatic graphs: the Anti-Ramsey problem introduced by Erdős et al. [5], rainbow connection problems introduced by Chartrand et al. [4], and heterochromatic subgraph problems, (see the surveys [6] [11] [9] ). This paper focuses on heterochromatic subgraph problems.

We denote by the number of components of a graph . Given an edge-colored graph and a color set , we define . For simplicity, we denote the graph by , and also denote by for a color .

Akbari & Alipour [1] and Suzuki [12] independently presented a necessary and sufficient condition for edge-colored graphs to have a heterochromatic spanning tree.

Theorem 1.1 (Akbari and Alipour [1], Suzuki [12]).

An edge-colored graph has a heterochromatic spanning tree if and only if

Suzuki [12] proved the following theorem by using Theorem 1.1.

Theorem 1.2 (Suzuki [12]).

An edge-colored complete graph of order n has a heterochromatic spanning tree if for any color .

The complete graph has edges, thus the condition of Theorem 1.2 is equivalent to that

We can regard as the probability of a color appearing in . The term “Heterochromatic” means that any color appears once or zero times. Thus, we can interpret Theorem 1.2 as saying that if each color probability is at most in then has a spanning tree such that each color probability is or in .

1.2 -Chromatic spanning trees and forests

The term “Heterochromatic” means that any color appears at most once. Suzuki [13] generalized “once” to a mapping from a given color set to the set of non-negative integers, and defined -chromatic graphs as follows.

Definition 1.3 (Suzuki [13]).

Let be an edge-colored graph. Let be a mapping from to . is said to be -chromatic if for any color .

Fig. 1 shows an example of an -chromatic spanning tree of an edge-colored graph. For the color set , a mapping is given as follows:

The left edge-colored graph has the right -chromatic spanning tree, where each color appears at most times.

Fig. 1: An -chromatic spanning tree of an edge-colored graph.

Suzuki [13] presented the following necessary and sufficient condition for edge-colored graphs to have an -chromatic spanning forest with exactly components.

Theorem 1.4 (Suzuki [13]).

Let be an edge-colored graph of order . Let be a mapping from to . Let be a positive integer such that . has an -chromatic spanning forest with exactly components if and only if

Suzuki [13] proved the following Theorem by using Theorem 1.4.

Theorem 1.5 (Suzuki [13]).

Let be an edge-colored graph of order . Let be a mapping from to . Let be a positive integer such that . If and

then has an -chromatic spanning forest with exactly components.

A heterochromatic graph is an -chromatic graph with for every color . Thus, these two theorems include Theorem 1.1 and Theorem 1.2. In this paper, we will further generalize these theorems and study color probability distributions of edge-colored complete graphs and its spanning trees.

2 Main results

In this paper, I propose a -chromatic graph as an edge-colored graph where each color appears at least times and at most times. I also present a necessary and sufficient condition for edge-colored graphs to have a -chromatic spanning forest with exactly components. (Theorem 2.2). Using this criterion, I show that an edge-colored complete graph has a spanning tree with a color probability distribution “similar” to that of (Theorem 2.4). Moreover, I conjecture that an edge-colored complete graph of order can be partitioned into edge-disjoint spanning trees such that each has a color probability distribution “similar” to that of (Conjecture 2.8).

2.1 -Chromatic spanning trees and forests

We begin with the definition of a -chromatic graph.

Definition 2.1.

Let be an edge-colored graph. Let and be mappings from to . is said to be -chromatic if for any color .

Fig. 2 shows a -chromatic spanning tree of an edge-colored graph. For the color set , mappings and are given as follows:

The left edge-colored graph has the right -chromatic spanning tree, where each color appears at least times and at most times.

Fig. 2: A -chromatic spanning tree of an edge-colored graph.

We will see more examples. First, we suppose that and are given as follows:

Then, the left edge-colored graph in Fig. 2 has no -chromatic spanning trees, because exceeds , the size of a spanning tree of the graph.

Next, in Fig. 3, we suppose that and are given as follows:

Then, in the left edge-colored graph, any subgraph having , , and edges colored with , , and , respectively, contains the right subgraph, which has a cycle. Thus, the left graph has no -chromatic spanning trees.

Fig. 3: The mapping forces us to use a cycle.

The following is the main theorem, which gives a necessary and sufficient condition for edge-colored graphs to have a -chromatic spanning tree as a corollary.

Theorem 2.2.

Let be an edge-colored graph of order . Let and be mappings from to such that for any . Let be a positive integer such that . has a -chromatic spanning forest with exactly components if and only if

This theorem is proved in Section 3.3. Note that the size of a spanning forest with exactly components of is . If has a -chromatic spanning forest with exactly components, then the size of the forest is at least . Thus, the condition is necessary.

We see the above last example again. Let be the left graph in Fig. 3. has no -chromatic spanning trees. Thus, by Theorem 2.2,

Actually, for , is the right graph in Fig. 3 and we have

We can prove the following theorem by using Theorem 2.2.

Theorem 2.3.

Let be an edge-colored graph of order . Let and be mappings from to . Let be a positive integer such that . If and

then has a -chromatic spanning forest with exactly components.

This theorem is proved in Section 3.4. Note that an -chromatic graph is a -chromatic graph with for any color , and for any since the number of components of any subgraph of a graph of order is at most . Thus, Theorem 2.2 and 2.3 include Theorem 1.4 and 1.5.

2.2 Color probability distributions of edge-colored graphs

We call the color probability of a color in an edge-colored graph . The color probability distribution of is the sequence of the color probabilities. Does a given edge-colored complete graph have a spanning tree with the same color probability distribution as that of ? Fig. 4 shows an example.

Fig. 4: An edge-colored complete graph having a spanning tree with the same color probability distribution as that of it.

Let and be the left and right graph in Fig. 4, respectively. Then,

Thus, both color probability distributions are (Fig. 5).

Fig. 5: Color probability distributions of the graphs in Fig. 4.

Fig. 6 shows another example. In the left complete graph of order has no spanning trees with the same color probability distribution as that of , because the number of colors in is exceeding the size of a spanning tree of .

Fig. 6: An edge-colored complete graph having no spanning trees with the same color probability distribution as that of it, but having a spanning tree with a color probability distribution similar to that of it.

In other words, for any spanning tree of ,

that is,

However, the right spanning tree has a color probability distribution similar to that of (see Fig. 7), in the sense that the following condition holds:

Fig. 7: Color probability distributions of the graphs in Fig. 6.

In general, let be an edge-colored complete graph of order , and set

Then, by Theorem 2.3, has a -chromatic spanning tree . By the definition 2.1, for any color . Thus, the following theorem holds.

Theorem 2.4.

Any edge-colored complete graph of order has a spanning tree with a color probability distribution similar to that of , that is, has a spanning tree such that

Remark 2.5.

Theorem 2.4 is equivalent to Theorem 2.3 with for any edge-colored complete graph of order .

Proof..

Theorem 2.4 follows from Theorem 2.3 by the above argument.

Let be an edge-colored graph of order . Let and be mappings from to . Suppose that Theorem 2.4 holds. Then, has a spanning tree with a color probability distribution similar to that of .

If ,, and satisfy the condition in Theorem 2.3, then we have

Hence, satisfies that for any color ,

Therefore, is a -chromatic spanning tree of . ∎

From Theorem 2.4, we can get the following theorem, proved in Section 3.5.

Theorem 2.6.

An edge-colored complete graph of order has a spanning tree with the same color probability distribution as that of if and only if is an integral multiple of for any color .

2.3 Spanning tree decomposition conjectures

In 1996, Brualdi and Hollingsworth [2] presented the following conjecture.

Conjecture 2.7 (Brualdi and Hollingsworth [2]).

A properly edge-colored complete graph with exactly colors can be partitioned into edge-disjoint heterochromatic spanning trees.

Brualdi and Hollingsworth [2] proved that a properly edge-colored complete graph with exactly colors has two edge-disjoint heterochromatic spanning trees. Krussel, Marshall, and Verrall [10] proved that the graph has three edge-disjoint heterochromatic spanning trees. Kaneko, Kano, and Suzuki [8] proved that a properly edge-colored complete graph (not necessary with exactly colors) has two edge-disjoint heterochromatic spanning trees. Akbari and Alipour [1] proved that an edge-colored complete graph (not necessary to be proper) of order has two edge-disjoint heterochromatic spanning trees if for any color . Carraher, Hartke, and Horn [3] proved that an edge-colored complete graph of order has at least edge-disjoint heterochromatic spanning trees if for any color . Horn [7] proved that there exist positive constants so that every properly edge-colored complete graph with exactly colors has at least edge-disjoint heterochromatic spanning trees.

Based on these previous results, I conjecture the following as a generalization of Conjecture 2.7.

Conjecture 2.8.

An edge-colored complete graph of order can be partitioned into edge-disjoint spanning trees such that each has a color probability distribution is similar to that of , that is, each satisfies that

Since for the complete graph of order , we have

Thus, this conjecture implies that can be partitioned into almost equal parts. Fig. 8 shows an edge-colored complete graph of order and its partition into three edge-disjoint spanning trees , , and . In this example,

Thus, is partitioned into three almost equal parts for each color , and each satisfies that

Hence, each has a color probability distribution similar to that of .

Fig. 8: An example of Conjecture 2.8.

By the same argument in the proof of Remark 2.5, we can show that Conjecture 2.8 is equivalent to the following proposition.

Conjecture 2.9.

Let be an edge-colored complete graph of order . Let and be mappings from to . If for any color ,

then can be partitioned into edge-disjoint -chromatic spanning trees.

3 Proofs

In this section, we will prove Theorem 2.2, Theorem 2.3, and Theorem 2.6. In order to prove Theorem 2.2, we will use Lemma 3.1 and 3.2, which will be proved in Section 3.1 and 3.2, respectively. In order to prove Theorem 2.3, we will use almost trivial Lemma 3.3, which was proved by Suzuki [13].

Lemma 3.1.

Let be an edge-colored graph of order . Let be a mapping from to . has a -chromatic forest if and only if

Note that, this lemma requires the forest neither to be a spanning forest nor to have a fixed number of components.

Lemma 3.2.

Let be an edge-colored graph of order . Let and be mappings from to such that for any . Let be a positive integer. has a -chromatic spanning forest with exactly components if and only if has both an -chromatic spanning forest of size at least with exactly components, and a -chromatic forest.

Note that, the -chromatic spanning forest and the -chromatic forest may be different in Lemma 3.2.

Lemma 3.3.

3.1 Proof of Lemma 3.1

Let be an edge-colored graph of order . Let be a mapping from to .

First, we prove the necessity. Suppose that has a -chromatic forest . By Definition 2.1, for any color . For any , the graph is a spanning forest of . Thus,

Next, we prove the sufficiency. Suppose that

Set . Then,

that is,

Thus, we have

Hence, by Theorem 1.4, has a -chromatic spanning forest with exactly components. By Definition 1.3, for any color . On the other hand, we have

Thus, for any color . Therefore, by Definition 2.1, is a -chromatic forest of .

3.2 Proof of Lemma 3.2

Let be an edge-colored graph of order . Let and be mappings from to such that for any . Let be a positive integer.

First, we prove the necessity. Suppose that has a -chromatic spanning forest with exactly components. By Definition 2.1, for any color . Thus, . Hence, is an -chromatic spanning forest of size at least with exactly components of . Since is a -chromatic forest, contains some -chromatic forest, which is also a -chromatic forest in .

Next, we prove the sufficiency. Suppose that has both an -chromatic spanning forest of size at least with exactly components, and a -chromatic forest . Let be an -chromatic spanning forest of size at least with exactly components of such that it has the maximum number of edges of .

We will prove that is the desired -chromatic spanning forest with exactly components of by contradiction.

Suppose that is not a -chromatic spanning forest with exactly components of . Then, since is -chromatic but not -chromatic, we may assume that for some color, say color , .

Since is -chromatic, . Thus, . Hence, . Let be an edge in . Adding the edge to , we consider the resulting graph denoted by . Since is -chromatic and , we have

Thus, is also an -chromatic spanning subgraph of .

If the edge connects two distinct components of in , then is an -chromatic spanning forest with exactly components of . Since is -chromatic, . Since , we have

Thus, . Let be an edge in . Then, we have

where denotes the graph . Hence, since is an -chromatic spanning forest of , is an -chromatic spanning forest of size at least with exactly components of . Recall that and . Then, , namely, has more edges of than , which is a contradiction to the maximality of .

Therefore, we may assume that the both endpoints of are contained in one component of . Then, and has exactly one cycle , which contains . Since has no cycles, has some edge . Then, is a forest and

Thus, since is an -chromatic spanning subgraph of , is an -chromatic spanning forest of size at least with exactly components of . Recall that and . Then, , namely, has more edges of than , which is a contradiction to the maximality of .

Consequently, is the desired -chromatic spanning forest with exactly components of .

3.3 Proof of Theorem 2.2

Let be an edge-colored graph of order . Let and be mappings from to such that for any . Let be a positive integer such that .

First, we prove the necessity. Suppose that has a -chromatic spanning forest with exactly components. Since is a -chromatic forest, contains some -chromatic forest. Thus, by Lemma 3.1, we have

On the other hand, since is a -chromatic spanning forest with exactly components of , is an -chromatic spanning forest with exactly components of . Thus, by Theorem 1.4, we have

Therefore,

Next, we prove the sufficiency. Suppose that

(1)

By (1), we have

Thus, by Theorem 1.4, has an -chromatic spanning forest with exactly components of . By our assumption that , we have

Thus, is an -chromatic spanning forest of size at least with exactly components.

On the other hand, by (1), we have

Thus, by Lemma 3.1, has a -chromatic forest.

Therefore, by Lemma 3.2, has a -chromatic spanning forest with exactly components.

3.4 Proof of Theorem 2.3

Let be an edge-colored graph of order . Let and be mappings from to . Let be a positive integer such that . Suppose that and

(2)

Then, since , we have

(3)

We will prove that has a -chromatic spanning forest with exactly components by contradiction.

Suppose that has no -chromatic spanning forests with exactly components. By (3) and our assumption, we can apply Theorem 2.2 to and we have

That is, or for some . We denote by .

Claim 1.
Proof..

First, we suppose that for some . Since for any color , .

By our assumption (2),