Question

Let \(G\) be a simple graph with \(n\) vertices, which is regular of degree \(d\). By considering the number of vertices that can be assigned the same colour, prove that \(\chi(G) \geq n /(n-d)\).

Short answer

The chromatic number \(\chi(G)\) of a given regular graph \(G\) with \(n\) vertices and degree \(d\) is greater than or equal to \(n/(n-d)\) as shown in the proof.

Step-by-step solution

Problem Interpretation

Given a simple graph \(G\) with \(n\) vertices, which is regular of degree \(d\). We need to prove that the chromatic number of \(G\), \(\chi(G)\), is greater or equal to \(n /(n-d)\).

Setting up the Proof

Suppose that \(G\) can be colored using \(c\) number of colors. As \(G\) is a regular graph of degree \(d\), every vertex in \(G\) is adjacent to exactly \(d\) others; hence each color class can have at most \(n-d\) vertices, because a vertex in one color class cannot be adjacent to a vertex of the same color.

Finding the Total Number of Vertices

We have at most \(n-d\) vertices for each color. Therefore, the total number of vertices can be at most \(c(n-d)\).

Comparing Total Number of Vertices with Actual Number of Vertices

Since the total number of vertices of the graph \(G\) is \(n\), it follows that \(c(n-d)\geq n\). Here, \(c\) represents the number of colors, which is exactly \(\chi(G)\), the chromatic number of \(G\).

Deriving the Inequality

From the inequality \(c(n-d)\geq n\), we can solve for \(c\), which gives us \(c\geq n/(n-d)\). So, ultimately we get \(\chi(G) \geq n/(n-d)\).

Key concepts

Graph Theory

Graph theory is a vital area of mathematics that studies the properties and applications of graphs. A graph in this context is a collection of points, known as vertices, and lines connecting them, called edges. This simple yet powerful mathematical concept can model complex relationships and interactions, making it widely applicable in various fields such as computer science, biology, and social sciences.

Understanding graph theory begins with familiarizing oneself with its basic terms and concepts. For instance, two vertices that an edge connects are said to be adjacent, and the degree of a vertex is the number of edges that connect to it. Special types of graphs exist, like simple graphs, where there are no loops or multiple edges between any two vertices. When tackling problems in graph theory, such as determining the chromatic number (a measure of how we can color a graph), one must grasp the underlying structure and interact with the graph's specific properties like regularity and degree of vertices.

Regular Graph

A regular graph has a uniform structure where each vertex has the same number of adjacent vertices, known as its degree. In graph theory, this consistent degree is a central feature of regular graphs and allows for the simplification of problems related to coloring and connectivity.

In the context of the provided exercise, a regular graph of degree d means every vertex is connected to exactly d other vertices. This regularity impacts how we can assign colors to the graph without violating coloring rules, specifically that no adjacent vertices can have the same color. When we look at a regular graph's colorability, we infer that the restrictions on color assignments derive directly from its degree, as it limits the number of vertices that can be the same color and thus can influence the graph's chromatic number.

Coloring Theorem

A coloring theorem in graph theory sets out to explain the minimum number of colors needed to color a graph's vertices, such that no two adjacent vertices have the same color; this minimal number is the graph's chromatic number, \( \chi(G) \). The provided exercise concerns a regular graph and challenges one to establish a lower bound for its chromatic number.

The theorem's beauty lies in its application to problems where the graph's regularity is known. The colors stand in for near-infinite potential applications, ranging from frequency assignments in telecommunications to scheduling problems. In examining this coloring problem, we apply the theorem to deduce that the minimum number of colors, hence the chromatic number, must be at least \( n/(n-d) \) for a regular graph with n vertices of degree d. This formula implies that as the degree d approaches the number of vertices n, the chromatic number increases, highlighting the complexity of coloring such graphs.