On the n-partite tournaments with exactly n-m+1 cycles of length m

When n-cycles in n-partite tournaments are longest cycles

An n-tournament is an orientation of a complete n-partite graph. It was proved by J.A. Bondy in 1976 that every strongly connected n-partite tournament has an n-cycle. We characterize strongly connected n-partite tournaments in which a longest cycle is of length n and, thus, settle a problem in L. Volkmann, Discrete Math. 245 (2002) 19-53.

On n-partite Tournaments with Unique n-cycle

An n-partite tournament is an orientation of a complete n-partite graph. An npartite tournament is a tournament, if it contains exactly one vertex in each partite set. Douglas, Proc. London Math. Soc. 21 (1970) 716-730, obtained a characterization of strongly connected tournaments with exactly one Hamilton cycle (i.e., n-cycle). For n ≥, we characterize strongly connected n-partite tournaments ...

Multipartite tournaments with small number of cycles

L. Volkmann, Discrete Math. 245 (2002) 19-53 posed the following question. Let 4 ≤ m ≤ n. Are there strong n-partite tournaments, which are not themselves tournaments, with exactly n − m + 1 cycles of length m? We answer this question in affirmative. We raise the following problem. Given m ∈ {3, 4, . . . , n}, find a characterization of strong n-partite tournaments having exactly n −m + 1 cycle...

$(m,n)$-algebraically compactness and $(m,n)$-pure injectivity

In this paper‎, ‎we introduce the notion of $(m,n)$-‎algebr‎aically compact modules as an analogue of algebraically‎ ‎compact modules and then we show that $(m,n)$-algebraically‎ ‎compactness‎ ‎and $(m,n)$-pure injectivity for modules coincide‎. ‎Moreover‎, ‎further characterizations of a‎ ‎$(m,n)$-pure injective module over a commutative ring are given‎.

Almost regular c-partite tournaments with c ≥ 8 contain an n-cycle through a given arc for 4 ≤ n ≤ c