Counterexamples to two conjectures on (1, 2)-domination in cubic graphs

Abstract

Let G be a cubic graph of order n. The induced cycles vertex number cind(G) is the largest size of a vertex set that induces a 2-regular subgraph of G. By gamma_1,2(G) we denote the (1,2)-domination number of G. Erves and Tepeh introduced the trilobite graphs T_n, which satisfy gamma_1,2(T_n) > cind(T_n). They stated two conjectures, the first of which says that every cubic graph G with cind(G) >= n/2 + 2 satisfies gamma_1,2(G) <= cind(G). The second says that a connected cubic graph G satisfies gamma_1,2(G) > cind(G) if and only if G is a trilobite. We show that both conjectures are false. A computer search finds counterexamples that are not trilobites already for n = 18, 20 and 22. We also construct an infinite family H(k) of order n = 20 + 4k. For every k >= 0 we prove that cind(H(k)) = n/2 + 2 and gamma_1,2(H(k)) = n/2 + 3. Hence both conjectures fail for infinitely many orders n.