Induced matching of cycle

Question

Definition: A graph $G$ is chordal if every induced cycle in $G$ has length 3, and is co-chordal if the complement graph $G^c$ is chordal.The co-chordal cover number, denoted $cochord (G)$, is the minimum number of co-chordal subgraphs required to cover the edges of $G$.

Definition: Induced matching in a graph $G$ is a matching which forms an induced subgraph of $G$, and that $indmatch (G)$ denotes the number of edges in a largest induced matching.

Suppose $G$ is cycle .

Is $cochord(G) \leq indmatch(G)+1$ true?

Answer 1

Yes. Let $C_n$ be the cycle on $n$ vertices. The size of a largest induced matching in $C_n$ is exactly $\lfloor \frac{n}{3} \rfloor$, since we can take at most every third edge of $C_n$. On the other hand, since $K_n$ minus a $3$-edge path is chordal, we have $cochord(C_n) \leq \lceil \frac{n}{3} \rceil$. Thus, $cochord(C_n) \leq indmatch(C_n)+1$, as required.