On the initial transition of graphs of Kirkman schedules by the partial team swap

Kirkman schedule is one of the typical single round-robin (abbrev. SRR) tournaments. The partial team swap (abbrev. PTS) is one of the typical procedures of changing from an SRR tournament to another SRR tournament, which is used in local search for solving the traveling tournament problem. An SRR of n teams (of even number) can be represented by a 1-factorization of the complete graph K_n$K_n$. It is known that the 1-factorization of any Kirkman schedule is “perfect” when n=p+1$n=p+1$ for prime numbers p, meaning that any pair of 1-factors in the 1-factorization forms a Hamilton cycle C_n$C_n$ in K_n$K_n$, called a 2-edge-colored Hamilton cycle. We are concerned with the cycle structure after applying the PTS to Kirkman schedules, that is, how a 2-edge-colored Hamilton cycle C_n$C_n$ is decomposed into two 2-edge-colored cycles of length 2d and n-2d$n-2d$, say, C_{2d}$C_{2d}$ and C_{n-2d}$C_{n-2d}$ for some number d\in [n/2]$d\in [n/2]$. We characterize the numbers d such that any cycle C_{2d}$C_{2d}$ is not generated by any PTS. Moreover, in case that a cycle C_{2d}$C_{2d}$ is generated, we show that the number of C_{2d}$C_{2d}$ for any d\ne n/4$d\ne n/4$ generated by any PTS is at most n-2$n-2$. For the case of d=n/4$d=n/4$ (i.e., C_{n/2}$C_{n/2}$), the number of C_{n/2}$C_{n/2}$ generated by any PTS is at most 2(n-2)$2(n-2)$, and there is some PTS to achieve the upper bound.