If $x$ is a vertex of a digraph $D$, then we denote by $d^+(x)$ and $d^-(x)$ the outdegree and the indegree of $x$, respectively. A digraph $D$ is called regular, if there is a number $p \in \mathbb{N}$ such that $d^+(x) = d^-(x) = p$ for all vertices $x$ of $D$. A $c$-partite tournament is an orientation of a complete $c$-partite graph. There are many results about directed cycles of a given length or of directed cycles with vertices from a given number of partite sets. The idea is now to combine the two properties. In this article, we examine in particular, whether $c$-partite tournaments with $r$ vertices in each partite set contain a cycle with exactly $r-1$ vertices of every partite set. In 1982, Beineke and Little [2] solved this problem for the regular case if $c = 2$. If $c \ge 3$, then we will show that a regular $c$-partite tournament with $r \ge 2$ vertices in each partite set contains a cycle with exactly $r-1$ vertices from each partite set, with the exception of the case that $c = 4$ and $r = 2$.
The complete tripartite graph $K_{n,n,n}$ has $3n^2$ edges. For any collection of positive integers $x_1,x_2,\dots ,x_m$ with $\sum _{i=1}^m x_i=3n^2$ and $x_i\ge 3$ for $1\le i\le m$, we exhibit an edge-disjoint decomposition of $K_{n,n,n}$ into closed trails (circuits) of lengths $x_1,x_2,\dots ,x_m$.
In this paper, the eigenvalue distribution of complex matrices with certain ray patterns is investigated. Cyclically real ray patterns and ray patterns that are signature similar to real sign patterns are characterized, and their eigenvalue distribution is discussed. Among other results, the following classes of ray patterns are characterized: ray patterns that require eigenvalues along a fixed line in the complex plane, ray patterns that require eigenvalues symmetric about a fixed line, and ray patterns that require eigenvalues to be in a half-plane. Finally, some generalizations and open questions related to eigenvalue distribution are mentioned.
We prove that for every number n ≥ 1, the n-iterated P3-path graph of G is isomorphic to G if and only if G is a collection of cycles, each of length at least 4. Hence, G is isomorphic to P3(G) if and only if G is a collection of cycles, each of length at least 4. Moreover, for k ≥ 4 we reduce the problem of characterizing graphs G such that Pk(G) ∼= G to graphs without cycles of length exceeding k.