Lower bound for the induced cycles vertex number in cubic graphs

Prove that every cubic graph G of order n satisfies c_ind(G) >= n/2, thereby establishing the conjectured lower bound for the induced cycles vertex number of cubic graphs.

Background

For a graph G, c_ind(G) is the largest cardinality of a vertex set inducing a 2-regular subgraph. Henning, Joos, Löwenstein, and Sasse established the general lower bound c_ind(G) >= (n+2)/4 for cubic graphs, while conjecturing that the bound can be doubled to n/2.

The paper constructs counterexamples to two conjectures relating c_ind(G) to the (1,2)-domination number, but explicitly notes that these examples do not challenge the stronger lower-bound conjecture c_ind(G) >= n/2. The conjecture remains unresolved in the class of all cubic graphs.

References

However, they conjectured in that the lower bound for cubic graphs is twice as large.

— Counterexamples to two conjectures on (1, 2)-domination in cubic graphs  (2608.17851 - Knor et al., 18 Aug 2026) in Conjecture 1, Section 1 (Introduction); abstract

In particular, the Goldberg family satisfies the conjecture of Henning et al. that $(G)\ge |V(G)|/2$ for every cubic graph $G$.

— On $1$-limited and $(1,2)$-domination in cubic graphs  (2610.01796 - Radić et al., 1 Oct 2026) in Conclusion