To prove that $\chi(G) \leq 4$ for a planar graph $G$ containing no $K_3$ (triangles) as a subgraph, we can rely on general properties of planar graphs and apply some specific reasoning about colorability.

Proof Outline

1. Basic Properties of Planar Graphs:

   • By Euler’s formula, for a connected planar graph, $V - E + F = 2$, where $V$ is the number of vertices, $E$ the number of edges, and $F$ the number of faces.

2. Degree Constraint in Triangle-Free Planar Graphs:

   • For any planar graph without triangles, each face is bounded by at least four edges (since there are no 3-cycles or triangles).
   • Applying Euler's formula and the property of planar graphs, one can show that the average degree of vertices in a triangle-free planar graph is less than $4$. Thus, such graphs are "sparse" in a specific sense.

3. Colorability Argument:

   • By the Four Color Theorem, any planar graph can be colored with at most 4 colors. However, the absence of $K_3$ as a subgraph implies that $G$ is "triangle-free," meaning it has no cycles of length 3.
   • According to Grötzsch's theorem, a planar triangle-free graph is actually 3-colorable. Therefore, for any planar graph without triangles, $\chi(G) \leq 3$.

4. Conclusion:

   • Since Grötzsch’s theorem applies here (given $G$ is planar and triangle-free), we conclude that $\chi(G) \leq 3$, which indeed satisfies $\chi(G) \leq 4$, as required by the problem.

Final Statement

Thus, we have shown that $\chi(G) \leq 4$ for any planar graph $G$ containing no $K_3$ as a subgraph. The stronger result, as noted, is that $\chi(G) \leq 3$ due to Grötzsch's theorem.

Would you like a deeper explanation on any step or a more detailed derivation of Euler's formula implications for triangle-free graphs?

Here are some related questions to deepen understanding:

1. How does Euler’s formula specifically apply to triangle-free planar graphs in terms of average degree?
2. Why does the Four Color Theorem apply to general planar graphs, not just triangle-free ones?
3. What makes Grötzsch’s theorem difficult to prove in comparison to other coloring results?
4. How does the absence of $K_3$ change the structural properties of a planar graph?
5. Can this approach be extended to other types of planar graphs with similar restrictions?

Tip: In planar graph problems, always consider if specific theorems like Euler’s or the Four Color Theorem can simplify the colorability or structural analysis.