Almost all almost regular c-partite tournaments with c⩾5 are vertex pancyclic

Multipartite tournaments: A survey

“…In the same year, Pan and Zhang [91] proved that almost all c-partite tournaments with c 5 and I (D) 1 are vertex pancyclic. As generalizations of the results in [117] for i g (D) 1, Yeo [167] showed that all sufficiently large c-partite tournaments with c 5 and bounded global irregularity are vertex pancyclic. Since the bounds given in Theorems 10.9 and 10.10 are probably not best possible, the following problems remain still open.…”

“…Using Theorems 2.1 and 3.3, it is a simple matter to verify that every multipartite tournament D with (D) (D) has also this property. Moreover, Tewes et al [117] even proved that every vertex of an almost regular c-partite tournament is contained in an m-cycle for each m ∈ {3, 4, . .…”

“…However, different examples have shown that this result is false for c = 3, even when D is regular. In addition, in [117], we have constructed almost regular 4-partite tournaments that are not vertex pancyclic.…”

“…Tewes et al [117] constructed some infinite families of almost regular 4-partite tournaments that are not vertex pancyclic. Thus, Theorem 10.6 as well as the next conjecture cannot be extended to almost regular 4-partite tournaments.…”

“…Conjecture 10.7 (Tewes et al [117]). An almost regular c-partite tournament with 5 c 7 is vertex pancyclic.…”