Question

Prove that if \(G\) is a cubic Hamiltonian graph, then \(\chi^{\prime}(G)=3\).

Short answer

Thus, according to the given proof, if \(G\) is a cubic Hamiltonian graph, then \(\chi^{\prime}(G) = 3\).

Step-by-step solution

Noting the important points

For a cubic graph, it's clear that \(\Delta(G) = 3\), as every vertex has degree 3. According to the Vizing's theorem, \(\Delta(G) \leq \(\chi^{\prime}(G)\) \leq \(\Delta(G) + 1\). Thus, the edge chromatic number of a cubic graph is either 3 or 4.

Observing the Hamiltonian Property

Since a Hamiltonian graph contains a Hamiltonian circuit, we can color the edges of the Hamiltonian cycle alternately with two colors, say color A and color B. This ensures that no two adjacent edges share the same color.

Coloring remaining edges in the graph

Since the graph is cubic and Hamiltonian, every vertex outside the Hamiltonian cycle must have one remaining edge. We can color these remaining edges with a third color, say color C. Thus, all edges of \(G\) are covered.

Concluding the proof

Since all the edges are colored and none of the vertex has two adjacent edges of same color and the total number of colors used is three, we conclude that \(\chi^{\prime}(G) = 3\).

Key concepts

Vizing's Theorem

Vizing's Theorem is a fundamental concept in graph theory that provides an important characterization of the edge chromatic number. It states that for a simple graph, the edge chromatic number, denoted as \( \chi'(G) \), falls between the maximum degree of the graph, \( \Delta(G) \), and \( \Delta(G) + 1 \). This means that any simple graph can have its edges colored using at most one more color than its maximum degree.
Let's imagine a cubic graph, where every vertex connects to exactly three others. According to Vizing's Theorem, the edge chromatic number \( \chi'(G) \) of this cubic graph could be either 3 or 4. This provides a starting point for proving further characteristics about the graph's coloring properties. By knowing this, it helps us in assessing the minimum number of colors we might need to properly color the edges without conflicts.

Hamiltonian Circuit

A Hamiltonian circuit, also known as a Hamiltonian cycle, is a cycle in a graph that visits every vertex exactly once before returning to the starting point. In the context of a cubic Hamiltonian graph, this circuit forms the backbone for exploring the graph's edge coloring properties.
To understand this further, consider a simple cube-like structure, where a Hamiltonian circuit provides a unique path that goes around all corners of the cube without retracing any steps. By taking advantage of this structure, we can start coloring the edges of the circuit.
Typically, we might choose two different colors to alternate the edges of the Hamiltonian cycle. This kind of coloring ensures that no two adjacent edges have the same color. In a cubic graph, this feature significantly helps in reducing the number of colors required for the rest of the structure.

Edge Chromatic Number

The edge chromatic number, denoted \( \chi'(G) \), is an important measure that indicates the smallest number of colors needed to color all the edges of a graph such that no two adjacent edges share the same color. In examining cubic Hamiltonian graphs, determining this number is central to understanding how efficiently such graphs can be colored.
A cubic graph, by definition, means every vertex connects with exactly three others. Now, when we incorporate the Hamiltonian aspect, it helps to simplify deriving the edge chromatic number. For a Hamiltonian cycle within a cubic graph, two colors might be sufficient to color all the edges of the cycle.
However, with vertices linked by additional edges outside this cycle, a third color steps in perfectly to ensure no color clashes at any vertex. Thus, combining the Hamiltonian circuit's coloring strategy with these external edges reveals that \( \chi'(G) = 3 \) for cubic Hamiltonian graphs. Ultimately, it aligns with Vizing's Theorem and confirms an optimal, efficient edge-coloring scheme for such specific graph structures.