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.