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. Our examples do not affect the conjecture of Henning et al. that cind(G) >= n/2 for every cubic graph G, which remains open.

• The paper refutes two Erveš–Tepeh conjectures through exhaustive searches of connected cubic graphs up to order 22 and the construction of an infinite family H(k).
• The family H(k), with order n = 20 + 4k, satisfies c_ind(H(k)) = n/2 + 2 and γ₁,₂(H(k)) = n/2 + 3, disproving both conjectures for every k ≥ 0.
• The results preserve the open conjecture c_ind(G) ≥ n/2 for cubic graphs while showing that (1,2)-domination cannot universally certify it, with counterexamples appearing from order 18.

This paper refutes two conjectures of Erveš and Tepeh concerning the relationship between the induced cycles vertex number $c_{\mathrm{ind}}(G)$ and the $(1,2)$-domination number $\gamma_{1,2}(G)$ of cubic graphs. The refutation is twofold: an exhaustive computer search over connected cubic graphs of order $n \le 22$ produces non-trilobite counterexamples already at $n = 18$, and an explicit infinite family $H(k)$ of order $n = 20 + 4k$ is constructed and analyzed, for which both conjectures fail for every $k \ge 0$. Importantly, all examples satisfy $c_{\mathrm{ind}}(G) \ge n/2 + 2$, so the underlying conjecture of Henning et al.\ that $c_{\mathrm{ind}}(G) \ge n/2$ for every cubic graph remains untouched.

Background and motivation

For a graph $(1,2)$0 of order $(1,2)$1, a set $(1,2)$2 is good if $(1,2)$3 is $(1,2)$4-regular, i.e., a disjoint union of cycles; the parameter $(1,2)$5 is the maximum size of a good set. Computing a largest induced $(1,2)$6-regular subgraph is NP-hard for every fixed $(1,2)$7, so structural lower bounds are of interest [Cardoso2007]. Henning, Joos, Löwenstein and Sasse proved $(1,2)$8 for every cubic graph and conjectured the sharp bound $(1,2)$9 [Henning2016]; this remains open in general, though it is known for restricted classes such as claw-free cubic graphs ($\gamma_{1,2}(G)$0) and connected cubic $\gamma_{1,2}(G)$1-chordal graphs other than three small exceptions ($\gamma_{1,2}(G)$2) [Henning2016b].

The relevance of $\gamma_{1,2}(G)$3-domination is that Fakhran et al.\ established $\gamma_{1,2}(G)$4 for every connected cubic graph [Fakhran2021]. Consequently, any cubic graph satisfying

$\gamma_{1,2}(G)$5

verifies the Henning et al.\ conjecture, since then $\gamma_{1,2}(G)$6. This inequality holds whenever some good set dominates $\gamma_{1,2}(G)$7. Erveš and Tepeh showed the route cannot work universally: their trilobite graphs $\gamma_{1,2}(G)$8 (three parallel strands joined by claw centres and capped at both ends by a $\gamma_{1,2}(G)$9 or triangle) satisfy $n \le 22$0 but $n \le 22$1. On the basis of trilobites and computational evidence they proposed two conjectures: first, that every cubic graph with $n \le 22$2 satisfies $n \le 22$3; second, that a connected cubic graph satisfies $n \le 22$4 if and only if it is a trilobite.

Computational counterexamples

An exhaustive search over all connected cubic graphs of order at most $n \le 22$5 shows that non-trilobite graphs violating $n \le 22$6 appear already at order $n \le 22$7: besides the trilobite itself there are three such graphs of order $n \le 22$8, four of order $n \le 22$9, and fifteen of order $n = 18$0. Several violate the threshold conjecture as well, and some graphs of order $n = 18$1 even attain $n = 18$2. This last point has a direct consequence: raising the hypothesis threshold from $n = 18$3 to $n = 18$4 would not repair the conjecture, since counterexamples exist exactly at that value. The counts also indicate that the exceptional class grows with $n = 18$5 rather than being confined to trilobites.

The family $n = 18$6

The family $n = 18$7 reuses the trilobite's claw-layer construction but replaces one end cap with a branching junction. For $n = 18$8, the graph consists of a $n = 18$9 cap $H(k)$0 on $H(k)$1, $H(k)$2 claw layers $H(k)$3 continuing three strands, a junction containing two triangles — $H(k)$4 merging strands $H(k)$5 and $H(k)$6, and $H(k)$7 splitting strand $H(k)$8 into two — and a second $H(k)$9 cap $n = 20 + 4k$0 attached via $n = 20 + 4k$1, $n = 20 + 4k$2, $n = 20 + 4k$3. The graph is connected, cubic, and has order $n = 20 + 4k$4. Its smallest member $n = 20 + 4k$5 coincides with one of the four non-trilobite counterexamples of order $n = 20 + 4k$6 found computationally; for $n = 20 + 4k$7 both parameters were confirmed by exhaustive enumeration over all $n = 20 + 4k$8 vertex subsets.

The $n = 20 + 4k$9-domination number

The proof that $k \ge 0$0 proceeds by local forcing lemmas combined with a discharging argument. Key constraints include: each cap contributes at least $k \ge 0$1 vertices to any $k \ge 0$2-dominating set, with refined structure when the contribution is minimal; each claw layer contributes at least $k \ge 0$3 vertex, and singleton layers force their neighbours to be large — no three consecutive light layers occur, and two consecutive singletons must lie on the same strand and force weight $k \ge 0$4 on adjacent layers. A discharging scheme transfers charge from heavy layers to maximal runs of one or two light layers, ensuring every layer ends with charge at least $k \ge 0$5, while the caps retain net contribution at least $k \ge 0$6. Analysis of the tail yields $k \ge 0$7, with case distinctions on $k \ge 0$8 sharpening the bound. Summing gives $k \ge 0$9; the matching upper bound is witnessed by an explicit dominating set built from a $c_{\mathrm{ind}}(G) \ge n/2 + 2$0-periodic strand pattern across alternating triple and singleton layers, together with a fixed $c_{\mathrm{ind}}(G) \ge n/2 + 2$1-vertex selection in the tail. One caveat: for $c_{\mathrm{ind}}(G) \ge n/2 + 2$2 the lower bound rests on exhaustive computation rather than the discharging argument, which applies only for $c_{\mathrm{ind}}(G) \ge n/2 + 2$3.

The induced cycles vertex number

The companion result $c_{\mathrm{ind}}(G) \ge n/2 + 2$4 uses a state-tracking argument instead of discharging. Each strand after each layer is assigned one of three states — absent, or present with one or two determined neighbours in $c_{\mathrm{ind}}(G) \ge n/2 + 2$5 — and only three aggregate states $c_{\mathrm{ind}}(G) \ge n/2 + 2$6 (with weights $c_{\mathrm{ind}}(G) \ge n/2 + 2$7) can occur, out of $c_{\mathrm{ind}}(G) \ge n/2 + 2$8 conceivable ones. The interface dynamics form a five-transition digraph, and the per-layer inequality $c_{\mathrm{ind}}(G) \ge n/2 + 2$9 telescopes to bound the total contribution of the claw layers. Cap analysis gives $c_{\mathrm{ind}}(G) \ge n/2$0, and tail analysis gives $c_{\mathrm{ind}}(G) \ge n/2$1, yielding $c_{\mathrm{ind}}(G) \ge n/2$2 for every good set. The lower bound is attained by an explicit good set that forms a single induced cycle of length $c_{\mathrm{ind}}(G) \ge n/2$3 winding through strands $c_{\mathrm{ind}}(G) \ge n/2$4 and $c_{\mathrm{ind}}(G) \ge n/2$5 and the junction; hence the longest induced cycle of $c_{\mathrm{ind}}(G) \ge n/2$6 also has length exactly $c_{\mathrm{ind}}(G) \ge n/2$7.

Consequences for the conjectures

Combining the two exact values, $c_{\mathrm{ind}}(G) \ge n/2$8 satisfies $c_{\mathrm{ind}}(G) \ge n/2$9 yet $(1,2)$00, and $(1,2)$01 is not a trilobite because its branching junction at the second end does not occur in any trilobite. Both conjectures of Erveš and Tepeh therefore fail simultaneously, for infinitely many orders $(1,2)$02, $(1,2)$03. Since every exhibited graph has $(1,2)$04, the original Henning et al.\ conjecture is unaffected; what is closed off is only the proposed route to it through $(1,2)$05. The paper leaves two questions open: whether every connected cubic graph with $(1,2)$06 arises from a trilobite or a member of $(1,2)$07 by bounded local modification, or whether sporadic exceptions grow without bound; and whether any finite threshold $(1,2)$08 makes the threshold conjecture true with $(1,2)$09 in place of $(1,2)$10 — the order-$(1,2)$11 examples show $(1,2)$12 would be required, and no finite $(1,2)$13 is known to suffice.

Conclusion

The paper disproves both conjectures linking $(1,2)$14-domination and induced cycle structure in cubic graphs, first by finite computation at orders $(1,2)$15–$(1,2)$16 and then by an infinite family with exactly determined parameters $(1,2)$17 and $(1,2)$18. The proofs combine discharging for the domination bound with a finite-state potential argument for the induced-cycle bound. The central conjecture $(1,2)$19 for cubic graphs remains open, and the results demonstrate that bounding it via $(1,2)$20 requires a more nuanced account of the graphs where the two parameters diverge.