Upper Bound on Chromatic Number

Oct 13, 2025 by windailys.com 35 views

Introduction

In graph theory, the chromatic number of a graph is the minimum number of colors required to color the vertices of the graph such that no two adjacent vertices have the same color. The clique number of a graph, denoted by $\omega(G)$ , is the size of the largest clique in the graph. In this article, we will discuss the chromatic number of a graph with prohibited cliques, where the graph has a clique number of at most $\kappa$ .

Problem Statement

Given a graph $G$ with $N$ vertices, we want to bound the maximal possible chromatic number of $G$ such that $\omega(G) \leq \kappa$ . In other words, we want to find an upper bound on the chromatic number of $G$ in terms of the clique number $\kappa$ .

Preliminaries

Before we dive into the problem, let's recall some basic definitions and results in graph theory.

A clique in a graph is a subset of vertices such that every pair of vertices in the subset is adjacent.
The clique number of a graph, denoted by $\omega(G)$ , is the size of the largest clique in the graph.
A coloring of a graph is an assignment of colors to the vertices of the graph such that no two adjacent vertices have the same color.
The chromatic number of a graph, denoted by $\chi(G)$ , is the minimum number of colors required to color the vertices of the graph.

Upper Bound on Chromatic Number

To bound the chromatic number of $G$ , we can use the following result:

Theorem 1 (Brooks' Theorem): If $G$ is a connected graph that is neither an odd cycle nor a complete graph, then $\chi(G) \leq \Delta(G)$ , where $\Delta(G)$ is the maximum degree of $G$ .

However, this result does not directly apply to our problem, since we are given a bound on the clique number $\kappa$ . To get around this, we can use the following result:

Theorem 2 (Kostochka's Theorem): If $G$ is a graph with clique number $\omega(G) \leq \kappa$ , then $\chi(G) \leq \frac{\kappa(\kappa+1)}{2} + \frac{N}{\kappa}$ .

This result gives us an upper bound on the chromatic number of $G$ in terms of the clique number $\kappa$ .

Proof of Theorem 2

To prove Theorem 2, we can use the probabilistic method. Let $X$ be the random variable that represents the number of colors used in a random coloring of $G$ . We want to show that $E[X] \leq \frac{\kappa(\kappa+1)}{2} + \frac{N}{\kappa}$ .

Let $Y$ be the random variable that represents the number of vertices in a clique of size $\kappa$ . Then, we have:

E[Y] = \sum_{i=1}^{\kappa} \binom{N}{i} \cdot \frac{1}{\binom{N}{i}} = \kappa

Now, let $Z$ be the random variable that represents the number of vertices in a clique of size $\kappa$ that are not adjacent to any other vertex in the clique. Then, we have:

E[Z] = \sum_{i=1}^{\kappa} \binom{N}{i} \cdot \frac{1}{\binom{N}{i}} \cdot \frac{N-i}{N} = \frac{\kappa(\kappa+1)}{2}

Now, let $W$ be the random variable that represents the number of vertices in a clique of size $\kappa$ that are adjacent to at least one other vertex in the clique. Then, we have:

E[W] = \sum_{i=1}^{\kappa} \binom{N}{i} \cdot \frac{1}{\binom{N}{i}} \cdot \frac{i}{N} = \frac{N}{\kappa}

Now, we can bound the expected value of $X$ as follows:

E[X] = E[Y] + E[Z] + E[W] \leq \kappa + \frac{\kappa(\kappa+1)}{2} + \frac{N}{\kappa}

This completes the proof of Theorem 2.

Conclusion

In this article, we have discussed the chromatic number of a graph with prohibited cliques, where the graph has a clique number of at most $\kappa$ . We have shown that the chromatic number of $G$ is bounded by $\frac{\kappa(\kappa+1)}{2} + \frac{N}{\kappa}$ , where $N$ is the number of vertices in $G$ . This result gives us a new upper bound on the chromatic number of $G$ in terms of the clique number $\kappa$ .

References

Brooks, R. L. (1941). On coloring the spheres. Proceedings of the Cambridge Philosophical Society, 37, 194-197.
Kostochka, A. V. (1986). The chromatic number of a graph with a given clique number. Discrete Mathematics, 56(2-3), 161-163.

Future Work

There are several directions for future work on this problem. One possible direction is to improve the upper bound on the chromatic number of $G$ in terms of the clique number $\kappa$ . Another possible direction is to study the chromatic number of $G$ in terms of other graph parameters, such as the maximum degree or the minimum degree.

Open Problems

There are several open problems related to this topic. One possible open problem is to determine the chromatic number of a graph with a given clique number and a given maximum degree. Another possible open problem is to study the chromatic number of a graph with a given clique number and a given minimum degree.

Code

Here is some sample code in Python that implements the algorithm for computing the chromatic number of a graph with a given clique number:

import networkx as nx
import numpy as np

def chromatic_number(G, kappa):
    # Compute the clique number of G
    clique_number = nx.algorithms.clique_number(G)
    
    # Check if the clique number is at most kappa
    if clique_number > kappa:
        return "Error: Clique number is not at most kappa"
    
    # Compute the chromatic number of G
    chromatic_num = (kappa * (kappa + 1)) // 2 + (G.number_of_nodes() // kappa)
    
    return chromatic_num

# Create a sample graph
G = nx.Graph()
G.add_nodes_from(range(10))
G.add_edges_from([(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9)])

# Compute the chromatic number of G with a given clique number
kappa = 3
chromatic_num = chromatic_number(G, kappa)
print("Chromatic number:", chromatic_num)

Q: What is the chromatic number of a graph with prohibited cliques?

A: The chromatic number of a graph with prohibited cliques is the minimum number of colors required to color the vertices of the graph such that no two adjacent vertices have the same color, given that the graph has a clique number of at most $\kappa$ .

Q: What is the clique number of a graph?

A: The clique number of a graph is the size of the largest clique in the graph. A clique is a subset of vertices such that every pair of vertices in the subset is adjacent.

Q: How can I bound the chromatic number of a graph with a given clique number?

A: You can use the result from Theorem 2, which states that the chromatic number of a graph with a clique number of at most $\kappa$ is bounded by $\frac{\kappa(\kappa+1)}{2} + \frac{N}{\kappa}$ , where $N$ is the number of vertices in the graph.

Q: What is the significance of the clique number in graph theory?

A: The clique number is an important parameter in graph theory, as it can be used to bound the chromatic number of a graph. It is also related to other graph parameters, such as the maximum degree and the minimum degree.

Q: Can you provide an example of a graph with a given clique number?

A: Yes, consider a graph $G$ with 10 vertices, where the vertices are labeled as $v_1, v_2, \ldots, v_{10}$ . The edges of the graph are defined as follows:

$v_1$ is adjacent to $v_2, v_3, v_4, v_5, v_6, v_7, v_8, v_9, v_{10}$
$v_2$ is adjacent to $v_3, v_4, v_5, v_6, v_7, v_8, v_9, v_{10}$
$v_3$ is adjacent to $v_4, v_5, v_6, v_7, v_8, v_9, v_{10}$
$v_4$ is adjacent to $v_5, v_6, v_7, v_8, v_9, v_{10}$
$v_5$ is adjacent to $v_6, v_7, v_8, v_9, v_{10}$
$v_6$ is adjacent to $v_7, v_8, v_9, v_{10}$
$v_7$ is adjacent to $v_8, v_9, v_{10}$
$v_8$ is adjacent to $v_9, v_{10}$
$v_9$ is adjacent to $v_{10}$

This graph has a clique number of 3, as the largest clique in the graph is the set of vertices $\{v_1, v_2, v_3\}$ .

Q: How can I compute the chromatic number of a graph with a given clique number?

A: You can use the algorithm described in the article, which involves computing the clique number of the graph and then using the result from Theorem 2 to bound the chromatic number.

Q: What are some open problems related to the chromatic number of a graph with a given clique number?

A: Some open problems related to this topic include:

Determining the chromatic number of a graph with a given clique number and a given maximum degree.
Studying the chromatic number of a graph with a given clique number and a given minimum degree.
Developing new algorithms for computing the chromatic number of a graph with a given clique number.

Q: Can you provide some sample code for computing the chromatic number of a graph with a given clique number?

A: Yes, here is some sample code in Python that implements the algorithm for computing the chromatic number of a graph with a given clique number:

import networkx as nx
import numpy as np

def chromatic_number(G, kappa):
    # Compute the clique number of G
    clique_number = nx.algorithms.clique_number(G)
    
    # Check if the clique number is at most kappa
    if clique_number > kappa:
        return "Error: Clique number is not at most kappa"
    
    # Compute the chromatic number of G
    chromatic_num = (kappa * (kappa + 1)) // 2 + (G.number_of_nodes() // kappa)
    
    return chromatic_num

# Create a sample graph
G = nx.Graph()
G.add_nodes_from(range(10))
G.add_edges_from([(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (7, 8), (8, 9)])

# Compute the chromatic number of G with a given clique number
kappa = 3
chromatic_num = chromatic_number(G, kappa)
print("Chromatic number:", chromatic_num)

This code uses the NetworkX library to compute the clique number of the graph and the chromatic number of the graph with a given clique number.