Question

What is the chromatic number of (i) each of the Platonic graphs? (ii) the complete tripartite graph \(K_{r, s t}\) ? (iii) the \(k\)-cube \(Q_{k}\) ?

Short answer

For the Platonic graphs, their chromatic numbers are 4, 2, 3, 3 and 4 respectively. For \(K_{r, s, t}\) type graphs, the chromatic number is 3. For \(Q_{k}\) type graphs, the chromatic number is always 2.

Step-by-step solution

Identify Platonic Graphs

The five Platonic graphs are associated with the five Platonic solids: Tetrahedron (the corresponding graph is \(K_{4}\)), Cube (\(Q_{3}\)), Octahedron (\(K_{2,2,2}\)), Dodecahedron, and Icosahedron. These graphs have chromatic numbers of 4, 2, 3, 3, and 4 respectively.

The complete tripartite graph \(K_{r,s,t}\)

In a complete tripartite graph with partitions of sizes r, s and t, adjacency occurs between vertices in different sets, not the ones in the same set. Thus, each set can be colored with a different color. Therefore, the chromatic number is 3.

The \(k\)-cube \(Q_{k}\)

For the k-cube, since it is a bipartite graph, it only requires 2 colors. Thus, the chromatic number of \( Q_{k} \) is 2, independent of k.

Key concepts

Platonic graphs

Platonic graphs are fascinating constructs rooted in the study of regular solids, which date back to ancient Greek geometry. These graphs are special because they derive from the five Platonic solids - figures where each face is an identical regular polygon, and the same number meets at every vertex. The five Platonic graphs are:

• Tetrahedron: with the graph representation as a complete graph, denoted as \(K_4\), and a chromatic number of 4. In this case, you need four colors because it is fully connected, meaning each vertex shares an edge with every other vertex.
• Cube: represents a 3-dimensional cube, rendered graphically as \(Q_3\), with a chromatic number of 2, reflecting its status as a bipartite graph.
• Octahedron: this can be represented as \(K_{2,2,2}\) with a chromatic number of 3, demonstrating its need for three colors to avoid adjacent vertices sharing the same hue.
• Dodecahedron and Icosahedron: these require 3 and 4 colors respectively, emphasizing their increasing complexity.The chromatic number essentially tells us the minimum number of colors needed to color the graph so that no two neighboring vertices share the same color. Knowing these properties makes Platonic graphs quite valuable in graph theory and practical applications like network topology.

complete tripartite graph

A complete tripartite graph, often symbolized as \(K_{r,s,t}\), is a type of graph that divides its vertices into three independent sets or partitions. Each vertex in one set is connected by an edge to every vertex in the other sets, but there are no edges between vertices within the same set. This structure makes it efficient in numerous applications, including networking and task scheduling.For this type of graph, the chromatic number is straightforward to determine. Since each partition group can have its unique color without worrying about conflicts within the set, the chromatic number typically matches the number of partition groups, making it 3. Complete tripartite graphs are extended from the more basic bipartite graphs and provide a framework for understanding complex relationships in more advanced problems. Keeping track of how these partitions are organized helps minimize potential conflicts in graph coloring, making them a pivotal example in understanding and applying graph theory basics.

chromatic number

The chromatic number of a graph is a fundamental concept that indicates how many colors are needed to color the vertices of the graph so that no two adjacent vertices share the same color. Imagine you have a map, and you want different regions (or nodes) that touch to have different colors; this is essentially what chromatic number helps determine.This number is not uniform and varies based on the graph's structural properties. A complete graph with \(n\) vertices, where each vertex connects to every other vertex, has a chromatic number of \(n\) because every vertex needs a different color. Conversely, bipartite graphs, being simpler, always have a chromatic number of 2, since it is possible to divide the graph into two sets where no two vertices within a set share an edge. Understanding chromatic numbers assists in various problem-solving scenarios where logical separation and categorization of components is essential. Real-world applications of chromatic numbers include anything from electronic circuit design to social networking analysis. These applications underscore the importance of determining efficient colorings to maintain order and optimize different systems.