Question

draw a graph with four odd vertices and at least one loop.
choose the correct graph below.
a.
a
/ \\
b c
\\ /
d
b.
a---b
| |
c---d
c.
a---c
|
b---d
d.
a---b---c---d
/
b
e. the described graph is impossible

Explanation:

In graph - theory, a loop at a vertex increases the degree of that vertex by 2 (since a loop starts and ends at the same vertex). The sum of the degrees of all vertices in a graph is an even number (by the Hand - shaking lemma: \(\sum_{v\in V}d(v)=2e\), where \(e\) is the number of edges).

If we have four odd - degree vertices (\(d(v_1)\equiv1\pmod{2}\), \(d(v_2)\equiv1\pmod{2}\), \(d(v_3)\equiv1\pmod{2}\), \(d(v_4)\equiv1\pmod{2}\)), the sum of their degrees \(\sum_{i = 1}^{4}d(v_i)\equiv4\equiv0\pmod{2}\). But if we add at least one loop (which contributes an even number to the degree sum), the sum of all vertex degrees in the graph will still be even.

However, let's check each option:

• Option A: All vertices have degree 2 (even).
• Option B: All vertices have degree 2 (even).
• Option C: If we assume no loops (since there is no visual indication of a loop in the standard graph - drawing sense for the given vertices in the option), and just count the number of edges incident to each vertex. But even if we consider the square - like structure (assuming no loops), vertices have even degrees.
• Option D: Vertices \(A\) and \(D\) have degree 1 (odd), \(B\) has degree 3 (odd), \(C\) has degree 3 (odd). But there is no loop in this graph.

Since we need a graph with four odd vertices and at least one loop, and no graph among A - D satisfies the loop condition (in the standard graph - drawing interpretation where a loop is a closed edge incident to a single vertex), the described graph is impossible.

Answer:

E. The described graph is impossible