Question

Prove that, if \(G\) is a disconnected simple graph, then its chromatic polynomial \(P_{G}(\mathrm{k}\) ) is the product of the chromatic polynomials of its components. What can you say about the degree of the lowest non-vanishing term?

Short answer

The chromatic polynomial of a disconnected simple graph \(G\), \(P_{G}(\mathrm{k}\) ), is indeed the product of the chromatic polynomials of its components, and the degree of the lowest non-vanishing term is equal to the maximum of the degrees of the lowest non-vanishing terms of its components.

Step-by-step solution

Definition Review

A Disconnected Graph has more than one Component, which are sub-graphs that are isolated from the rest. The Chromatic Polynomial of a graph, \(P_{G}(k)\), denotes the number of ways to colour the vertices of the graph \(G\) with \(k\) or less colours, such that no two adjacent vertices share the same colour.

Identifying Components

Given \(G\) is Disconnected, identify its components, say \(G1, G2, ..., Gn\). They are all simple graphs and are isolated from each other.

Chromatic Polynomial of Components

Compute the chromatic polynomials for each component \(G1, G2, ..., Gn\) as \(P_{G1}(k), P_{G2}(k), ..., P_{Gn}(k)\) respectively.

Product of Chromatic Polynomials

Once the chromatic polynomial for each component has been found, multiply these together to get the chromatic polynomial of the entire graph \(G\), i.e., \(P_{G}(k) = P_{G1}(k) \times P_{G2}(k) \times ... \times P_{Gn}(k)\)

Degree of Lowest Non-Vanishing Term

The degree of the lowest non-vanishing term for the chromatic polynomial of a graph \(G\) corresponds to the minimal number of colours needed to properly colour the graph. For a disconnected graph, this will be equal to the maximum of the degrees of the lowest non-vanishing terms of its components.

Key concepts

Disconnected Graph

A disconnected graph is a unique type of graph structure characterized by the presence of more than one distinct sub-graph, or component, which are not connected by any edges. In simpler terms, these components within the graph exist in isolation from each other.

Understanding disconnected graphs is essential when exploring topics like graph theory and chromatic polynomials. The graph is called "simple" as it contains no loops or multiple edges.

• The concept of disconnected graphs is pivotal for calculations, such as finding the chromatic polynomial—a function that counts the number of ways a graph's vertices can be colored using a maximum number of colors without two adjacent vertices sharing the same color.
• With disconnected graphs, each component must be considered independently when calculating the chromatic polynomial.

By focusing on each component, the challenge of coloring becomes a series of smaller, separate tasks, which provides a more manageable approach to solving these types of graph problems.

Graph Components

In graph theory, each part of a disconnected graph is known as a component. A component is essentially a sub-graph that is isolated from others within a disconnected graph. These are essentially separate entities that do not share any edges with other sub-graphs in this structure.

This makes the task of computing the chromatic polynomial more straightforward, as it can be resolved through simpler calculations. Understanding graph components is fundamental, because:

• Each component can be thought of as a simple graph by itself.
• The chromatic polynomial of a disconnected graph is the product of the chromatic polynomials of its components.

When you focus on these components, you'll compute the chromatic polynomial of each separately, then multiply these polynomials to get the overall chromatic polynomial for the entire disconnected graph. This method emphasizes the modularity of understanding and analyzing complex graph structures.

Vertex Coloring

Vertex coloring is a technique used in graph theory where each vertex within a graph is assigned a color, following specific rules.

The main rule: no two adjacent vertices, meaning vertices connected directly by an edge, can share the same color. This constraint ensures that the graph is "properly colored." The objective often involves finding the minimum number of colors needed to achieve this proper coloring.

• In a chromatic polynomial, this concept articulates how many different ways a graph's vertices can be colored for any given number of colors.
• In disconnected graphs, because components are independent sub-graphs, each can be colored separately using the minimal total number of colors required by any one of them.

By breaking down vertex coloring into components, it becomes evident that the degree of the lowest non-vanishing term for the chromatic polynomial is determined by the component that requires the most colors. This is why understanding vertex coloring is crucial for graph analysis, particularly in the context of calculating chromatic polynomials for disconnected graphs.