Denote by χ(G) the chromatic number of a graph G. Let G be a graph with no 3-cycle. Let e1 and e2 be two distinct edges of G with a common end-vertex. Let H = G − {e1, e2} be the spanning subgraph of G obtained from G by deleting e1 and e2. Prove that
χ(G) − 1 ≤ χ(H) ≤ χ(G).

To prove the given inequality \(\chi(G) - 1 \leq \chi(H) \leq \chi(G)\), we need to show two things:

1. \(\chi(H) \leq \chi(G)\)
2. \(\chi(G) - 1 \leq \chi(H)\)

Let's start by understanding what each part means in the context of graph theory:

- \(\chi(G)\) is the chromatic number of \(G\), i.e., the minimum number of colors needed to color the vertices of \(G\) such that no two adjacent vertices share the same color.
- \(H = G - \{e1, e2\}\) is the graph obtained by deleting the edges \(e1\) and \(e2\) from \(G\).

### Part 1: \(\chi(H) \leq \chi(G)\)

Since \(H\) is obtained from \(G\) by removing edges, the chromatic number of \(H\) cannot be greater than that of \(G\). This is because removing edges can only reduce or maintain the number of required colors; it cannot increase it. Thus, if we can color \(G\) with \(\chi(G)\) colors, we can also color \(H\) with at most \(\chi(G)\) colors:

\[
\chi(H) \leq \chi(G)
\]

### Part 2: \(\chi(G) - 1 \leq \chi(H)\)

For this part, consider the structure of the graph \(G\) and \(H\):

- \(G\) has no 3-cycles, which implies \(G\) is triangle-free.
- \(e1\) and \(e2\) are two distinct edges with a common end-vertex. Let this common vertex be \(v\), and let the other vertices connected by \(e1\) and \(e2\) be \(u\) and \(w\), respectively. Thus, \(e1 = (v, u)\) and \(e2 = (v, w)\).

When we remove \(e1\) and \(e2\) to form \(H\), the vertex \(v\) will have fewer connections in \(H\) than in \(G\). This potentially reduces the constraints on coloring \(v\), but not necessarily the overall chromatic number of \(H\). 

Let's argue by contradiction. Assume \(\chi(H) < \chi(G) - 1\). This would imply that we can color \(H\) with \(\chi(G) - 2\) colors. But since \(H\) is \(G\) minus two edges, if we could color \(H\) with \(\chi(G) - 2\) colors, we could extend this coloring to \(G\) by appropriately adjusting the colors of vertices \(u\), \(v\), and \(w\) (since adding two edges back cannot force us to require two additional colors). This would contradict the definition of \(\chi(G)\) as the minimum number of colors needed.

Therefore, it must be that removing the two edges reduces the chromatic number by at most one:

\[
\chi(G) - 1 \leq \chi(H)
\]

Combining these results, we get the desired inequality:

\[
\chi(G) - 1 \leq \chi(H) \leq \chi(G)
\]

Would you like more details or have any questions?

Here are 5 questions you might want to ask next:
1. How is the chromatic number of a graph determined?
2. What are some common techniques for coloring a graph?
3. Can you provide an example of finding the chromatic number of a specific graph?
4. What is the significance of the chromatic number in graph theory?
5. Are there any famous theorems related to graph coloring?

Tip: When solving graph theory problems, always consider how modifying the graph (e.g., adding or removing edges) affects properties like chromatic number and connectedness.

Learn how to prove the inequality involving chromatic numbers of graphs without 3-cycles. Understand the concept of chromatic number, spanning subgraphs, and graph coloring techniques. Suitable for undergraduate level studies in graph theory.

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