Determine anti-Ramsey numbers for the opposite parity

Determine Ar(n,{C2k∪K2,C2k+1}).

Background

The paper states that the analogous anti-Ramsey question for the other parity remains open after discussing the odd-cycle case.

References

However, the analogous question for the other parity remains open. Problem 10.30. Determine Ar(n, {C2k U K2, C2k+1}).

— Monochromatic graph decompositions and monochromatic piercing inspired by anti-Ramsey colorings  (2501.09342 - Caro et al., 16 Jan 2025) in Problem 10.30, Section 10.2