To prove that every outerplanar graph has a chromatic number at most 3, we can proceed by using properties of outerplanar graphs and graph coloring concepts. Here’s a step-by-step explanation:

1. Outerplanar Graph Properties

• An outerplanar graph is a graph that can be embedded in the plane such that all vertices belong to the outer face of the embedding.
• Outerplanar graphs are a subset of planar graphs, and they have specific properties:
  • They do not contain $K_4$ (the complete graph on 4 vertices) as a minor.
  • They do not contain $K_{2,3}$ (the complete bipartite graph with partitions of size 2 and 3) as a minor.
• Importantly, outerplanar graphs are known to be 2-degenerate, which means they can be reduced to an empty graph by repeatedly removing vertices of degree 2 or less.

2. Chromatic Number and 2-Degeneracy

• A graph with a maximum degeneracy of $k$ can be colored with $k+1$ colors using a greedy coloring algorithm.
• Since outerplanar graphs are 2-degenerate, we can conclude that their chromatic number is at most $2 + 1 = 3$.

3. Proof Outline Using Greedy Coloring

• We can color an outerplanar graph by removing vertices one by one (due to its 2-degeneracy property) and coloring each vertex as we add it back.
• Each vertex, when re-added, has at most 2 neighbors already colored (since it was removed as a vertex of degree 2 or less).
• By using three colors, we can always assign a color to the vertex being re-added that differs from the colors of its neighbors.

4. Conclusion

• Therefore, every outerplanar graph can be colored with at most 3 colors, meaning its chromatic number is at most 3.

This completes the proof.

Would you like further details on any part of this explanation? Here are some related questions you might find interesting:

1. What is the significance of a graph being 2-degenerate in terms of vertex removal?
2. How does a 2-degenerate property differ from planarity in general graphs?
3. Can this approach be extended to graphs that are not outerplanar but planar?
4. What is the chromatic number of a general planar graph, and how does it relate to outerplanar graphs?
5. How does the absence of $K_4$ and $K_{2,3}$ in outerplanar graphs impact their structure?

Tip: When studying graph properties, always check for minor-closed conditions, as they often simplify proofs and lead to conclusions about the graph's coloring and structure.