Determine the maximum number of alternating (4t+2)-cycles

Determine the maximum number of copies of an alternating (4t+2)-cycle, including the alternating 6-cycle, in a red-blue complete graph, and establish the asymptotically extremal colourings.

Background

The paper proves sharp results for alternating cycles whose lengths are divisible by four. It observes that the corresponding argument does not extend to lengths congruent to two modulo four: balanced partitioned graphs contain no such cycles, whereas random colourings produce the expected order of magnitude. The authors explicitly state that they suspect the random-colouring value is optimal and list the alternating 6-cycle and the general (4t+2)-cycle as open semi-inducibility cases.

References

In contrast, we believe that the bound in Theorem~\ref{th:2kCyclesBound} is not achievable for alternating $(4t+2)$-cycles. Indeed, note that a partitioned graph does not contain any alternating $(4t+2)$-cycles and hence every alternating $(4t+1)$-walk is wasted'. On the other hand, it is straightforward to see that the uniformly random red-blue complete graph contains $\frac{1}{4t+2}(\frac{n}{2})^{4t+2} + O(n^{4t+1})$ alternating $(4t+2)$-cycles with high probability, and so precisely half of the alternating $(4t+1)$-walks arewasted'. We suspect that this is optimal.

— The semi-inducibility problem  (2501.09842 - Basit et al., 16 Jan 2025) in Section 5, “Alternating even cycles” and Section 8, “Concluding remarks”