Question

Let \(G\) be an undirected graph, which is a path on 8 vertices. The number of matchings in \(G\) is ______. (answer in integer)

Hint:

The number of matchings in a path on \(n\) vertices follows the Fibonacci-style recurrence \(m(n) = m(n-1) + m(n-2)\), with \(m(1)=1\) and \(m(2)=2\).

Answer 1

Correct Answer: 34

Step 1: Define a matching and set up notation.
A matching in a graph is a set of edges, possibly empty, such that no two edges share a vertex. Let \(m(n)\) be the number of matchings (including the empty matching) in the path graph \(P_n\) on \(n\) vertices \(v_1, v_2, \ldots, v_n\) with edges \((v_1,v_2), (v_2,v_3), \ldots, (v_{n-1}, v_n)\).

Step 2: Derive a recurrence by looking at the last vertex \(v_n\).
Either \(v_n\) is unmatched, in which case the rest of the matching is any matching of the path on the remaining \(n-1\) vertices \(v_1, \ldots, v_{n-1}\), giving \(m(n-1)\) possibilities. Or \(v_n\) is matched, and since \(v_n\) has only one possible edge in a path, namely \((v_{n-1}, v_n)\), it must be matched using that edge; then \(v_{n-1}\) is used up, and the rest of the matching is any matching of the remaining path on \(v_1, \ldots, v_{n-2}\), giving \(m(n-2)\) possibilities. So \(m(n) = m(n-1) + m(n-2)\).

Step 3: Base cases.
\(m(1) = 1\), a single vertex has no edges, so only the empty matching exists.
\(m(2) = 2\), a single edge, so matchings are the empty set and the set containing that one edge.

Step 4: Build up the sequence using the recurrence up to \(n=8\).
\(m(1)=1\)
\(m(2)=2\)
\(m(3)=m(2)+m(1)=2+1=3\)
\(m(4)=m(3)+m(2)=3+2=5\)
\(m(5)=m(4)+m(3)=5+3=8\)
\(m(6)=m(5)+m(4)=8+5=13\)
\(m(7)=m(6)+m(5)=13+8=21\)
\(m(8)=m(7)+m(6)=21+13=34\)

Step 5: Sanity check on \(m(3)=3\) and \(m(4)=5\) by direct listing. For \(P_3\) with edges \(e_1=(v_1,v_2)\), \(e_2=(v_2,v_3)\), the matchings are \(\{\}, \{e_1\}, \{e_2\}\), a total of 3, matching the formula. For \(P_4\) with edges \(e_1,e_2,e_3\), the matchings are \(\{\}, \{e_1\}, \{e_2\}, \{e_3\}, \{e_1,e_3\}\), a total of 5, again matching the formula. This confirms the recurrence is being applied correctly.

\[ \boxed{34} \]