Abstract

Max-Cut problem is one of the classical problems in graph theory and has been widely studied in recent years. Maximum colored cut problem is a more general problem, which is to find a bipartition of a given edge-colored graph maximizing the number of colors in edges going across the bipartition. In this work, we gave some lower bounds on maximum colored cuts in edge-colored complete graphs containing no rainbow triangles or properly colored 4-cycles.

1. Introduction

One of the classical partition problems is the well-known Max-Cut problem: given a graph , find a partition of maximizing the number of edges going across and . It has been a very active research subject in both combinatorics and computer science in the last 30 years. For a graph , let be the maximum number of edges in a cut of . For an integer , let denote the minimum value of , as ranges over all graphs with edges. A simple probabilistic argument or a greedy algorithm shows that . Answering a question of Erdos, Edwards [1, 2] in 1973 proved thatand this is right as evidenced by complete graph with an odd number of vertices.

In this work, we pay our attention to maximum colored cuts in edge-colored graphs: given a simple graph with an edge coloring , and for each , , find a partition of maximizing the number of colors of edges going across and . Faria et al. [3] proposed this problem and studied its complexity. Let . This problem converts to Max-Cut problem when .

For an edge-colored graph , we use to denote the spanning subgraph of induced by the edges of color . We use to denote the set of colors appearing on . For each , we also use to denote the color appearing on . For any two disjoint subgraphs and of , we use to denote the set of colors appearing on the edges between and . If , then we write instead of . For any two disjoint subsets and of , we use as shorthand for , where denote the subgraph of induced by . We say that is properly colored if each pair of adjacent edges in are assigned distinct colors. We say that is monochromatic if . And we say that is rainbow if all edges of are assigned different colors. Let be a complete graph with vertices. A rainbow is called a rainbow triangle. For a vertex , the color degree of , denoted by , is the number of different colors appearing on the edges incident with .

For an edge-colored graph with colors, let be the maximum number of colors in a cut of . Let denote the minimum value of , as ranges over all edge-colored graphs with colors. To distinguish from Max-Cut in graphs, we first need to forbid the following situation: is close to . If , then is close to . In addition, let be an edge-colored complete graph with colors such that a subgraph is rainbow and the remaining edges are monochromatic. Then, it is easy to verify that although . In these cases, we can give the lower bounds of the maximum colored cuts with the help of the results of the maximum cuts in graphs. To avoid these situations, we in this paper just discuss maximum colored cuts in edge-colored complete graphs, which contains no rainbow triangles or properly colored 4-cycles.

Using the probabilistic approach, we first give a lower bound of by constraining .

Theorem 1. Letbe an edge-colored graph with. For each color, if, then.

The following four results discuss the lower bound of , as is an edge-colored complete graph containing no rainbow triangles, properly colored 4-cycles, or monochromatic 4-cycles.

Theorem 2. Letbe an edge-colored complete graph with. Ifcontains no rainbow triangles or properly colored 4-cycles, then.

Theorem 3. Letbe an edge-colored complete graph with. If contains no properly colored 4-cycles, then.

Theorem 4. Letbe an edge-colored complete graph withvertices andcolors. Ifcontains no rainbow triangles, then there is a constant, such that.

Theorem 5. Letbe an edge-colored complete graph with. Ifcontains no rainbow triangles or monochromatic 4-cycles, then.
In the remainder of this paper, some useful lemmas are given in Section 2, the proofs of Theorems 1–3 are given in Section 3, and the proofs of Theorem 4 and Theorem 5 are given in Section 4. Finally, we enumerate some open problems.

2. Preliminaries

We first state a useful lemma on the structure of edge-colored complete graphs containing no properly colored 4-cycle.

Lemma 1 (see Martin et al. [4]). Letbe an edge-colored complete graph containing no properly colored 4-cycle. Then for each color, contains a dominating vertex.

The following fundamental result on the existence of properly colored cycles in colored graphs plays a key role in our proofs.

Lemma 2 (see Grossman and Häggkvist [5] and Yeo [6]). Letbe an edge-colored graph containing no properly colored cycles. Thencontains a vertexsuch that no component ofis joined towith edges of more than one color.
For an edge-colored graph , a partition of is called a Gallai partition if and for .

Lemma 3 (see Gallai [7]). Letbe an edge-colored. Ifcontains no rainbow triangles, thenhas a Gallai partition.
A graph has vertices , let be the degree of for . The sequence is called a degree sequence of . The following lemma, proved by Alon et al. [8], gives a lower bound of with respect to degree sequence of a sparse graph .

Lemma 4 (see Alon et al. [8]). There exists an absolute constantsuch that, for every constant, there is a constantwith the following property. Letbe a graph withvertices,edges and degree sequence. Suppose that the induced subgraph on any set ofvertices all of which have a common neighbor contains at mostedges. Then,

3. Proofs of Theorems 1–3

Proof of Theorem 1. Let be a random partition of by placing each into or , independently, with probability 1/2. For each color , let be the indicator random variable of the event that there exists an edge colored such that , or , . Recall that . Then, there is a vertex , which incidents with at least edges-colored . Let denote the set of colors appearing on the edges between and . Then,By the linearity of expectation,Thus, there exists a partition such that .

Proof of Theorem 2. Let be an edge-colored complete graph containing no rainbow triangles or properly colored 4-cycles. Li et al [9] gave an observation, which shows that this graph contains no properly colored cycles. For the sake of completeness, we recall the proof. By contradiction, suppose that is a properly colored cycle of minimal length in . Without loss of generality, assume that and , then . Since otherwise, or would be a properly colored cycle shorter than , a contradiction. Similarly, . Since otherwise, or would be a properly colored cycle shorter than , a contradiction too. Assume that . Recall that is not a properly colored cycle. Hence, or 3. If , then would be a properly colored cycle, a contradiction, so . Similarly, since is not a properly colored cycle. Hence, or 3. If , then would be a properly colored cycle, So . Thus, would be a properly colored cycle.
Note that contains no properly colored cycle. Then, by Lemma 2, there is a such that . Note that contains no properly colored cycle. There is a such that . Keep on doing this, we obtain a vertex sequence of and a graph sequence , such that for .
Now we place into and place into . Thus, . Then, we place into and place into , Thus, and . The rest vertices can be placed in this way. Then, we have for . Note that . Hence, we can obtain a partition such that each edges colored fall into .

Proof of Theorem 3. Let be an edge-colored complete graph containing no properly colored 4-cycle. We get the result easily if . So, we suppose that . If for each color , , then by Theorem 1, we have . We are done. Now, we suppose that there exists a color such that is a match. By Lemma 1, contains a dominating vertex. So is an edge. Assume that . If , then we are done. So we suppose that , which implies that there is an edge in , say , colored and . Thus, would be a properly colored 4-cycle, a contradiction.

4. Proofs of Theorems 4 and 5

Proof of Theorem 4. Let be an edge-colored containing no rainbow triangles. By Lemma 3, has a Gallai partition . Without loss of generality, suppose that . Let be a subgraph of induced by . Then, has a Gallai partition . Continuing this process, without loss of generality, we can get a that and . Let . By the definition of Gallai partition, we have . Let . Then, there exists a vertex, say , such that . Let . Similarly, there exists a vertex, say , such that . Continuing this process, for each , there exists such that . We get a vertex sequence . Now we construct a rainbow subgraph of such that . Let be a graph obtained from by keeping one arbitrary edge from each color. Then, we have . Note that contains no triangles. Let be a degree sequence of . Let denotes the number of neighbors of with . Note that . Then,By Lemma 4, we complete the proof.

Proof of Theorem 5. Let be an edge-colored complete graph containing no rainbow triangles and monochromatic 4-cycle. We claim that there is at most 1 color , such that . Otherwise, let , such that and . Let be colored , and let be colored . Obviously, and are not incident, as contains no rainbow triangles, which implies that . Thus, would be a monochromatic 4-cycle, a contradiction. So, there is at most 1 color in that the edges colored and the color is a match. Let be the subgraph of by deleting the match. By Theorem 1, we have .

5. Open Problems

In this section, we naturally proposed the following two problems on maximum colored cuts in edge-colored complete graphs.

Problem 1. What is the smallestsuch that each edge-colored complete graphwithcolors containing no rainbow triangles admits a partitionsatisfying?

Problem 2. What is the smallestsuch that each edge-colored complete graphwithcolors containing no properly colored 4-cycles admits a partitionsatisfying?

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

This work was supported by Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 21JK0994), Natural Science Basic Research Program of Shaanxi (Program No. 2022JQ-026), and YDBK 2019-71.