In this paper, we compute competition indices and periods of multipartite tournaments. We first show that the competition period of an acyclic digraph $D$ is one and $ζ(D) +1$ is a sharp upper bound of the competition index of $D$ where $ζ(D)$ is the sink elimination index of $D$. Then we prove that, especially, for an acyclic $k$-partite tournament $D$, the competition index of $D$ is $ζ(D)$ or $ζ(D) +1$ for an integer $k \ge 3$. By developing useful tools to create infinitely many directed walks in a certain regular pattern from given directed walks, we show that the competition period of a multipartite tournament with sinks and directed cycles is at most three. We also prove that the competition index of a primitive digraph does not exceed its exponent.
Decomposing a digraph into subdigraphs with a fixed structure or property is a classical problem in graph theory and a useful tool in a number of applications of networks and communication. A digraph is strongly connected if it contains a directed path from each vertex to all others. In this paper we consider multipartite tournaments, and we study the existence of a partition of a multipartite tournament with $c$ partite sets into strongly connected $c$-tournaments. This is a continuation of the study started in 1999 by Volkmann of the existence of strongly connected subtournaments in multipar…
For an integer $k \ge 2$, let $A$ be a Boolean block matrix with blocks $A_{ij}$ for $1 \le i,j \le k$ such that $A_{ii}$ is a zero matrix and $A_{ij}+A_{ji}^T$ is a matrix with all elements $1$ but not both corresponding elements of $A_{ij}$ and $A_{ji}^T$ equal to $1$ for $i \neq j$. Jung~{\em et al.} [Competition periods of multipartite tournaments. {\it Linear and Multilinear Algebra}, https://doi.org/10.1080/03081087.2022.2038057] studied the matrix sequence $\{A^m(A^T)^m\}_{m=1}^{\infty}$. This paper, which is a natural extension of the above paper and was initiated by the observation th…
The niche graph of a digraph $D$ has $V(D)$ as the vertex set and an edge $uv$ if and only if $(u,w) \in A(D)$ and $(v,w) \in A(D)$, or $(w,u) \in A(D)$ and $(w,v) \in A(D)$ for some $w \in V(D)$. The notion of niche graph was introduced by Cable et al. (1989) as a variant of competition graph. If a graph is the niche graph of a digraph $D$, it is said to be niche-realizable through $D$. If a graph $G$ is niche-realizable through a $k$-partite tournament for an integer $k \ge 2$, then we say that the pair $(G, k)$ is niche-realizable. Bowser et al. (1999) studied the graphs that are niche-real…
Following on the King Chicken Theorems originally proved by Maurer, we examine the idea of multiple flocks of chickens by bringing the chickens from tournaments to multipartite tournaments. As Kings have already been studied in multipartite settings, notably by Koh-Tan and Petrovic-Thomassen, we examine a new type of chicken more suited than Kings for these multipartite graphs: Dukes. We define an M-Duke to be a vertex from which any vertex in a different partite set is accessible by a directed path of length at most M. In analogy with Maurer's paper, we prove various structural results regard…
Given a tournament $T$, a module of $T$ is a subset $X$ of $V(T)$ such that for $x, y\in X$ and $v\in V(T)\setminus X$, $(x,v)\in A(T)$ if and only if $(y,v)\in A(T)$. The trivial modules of $T$ are $\emptyset$, $\{u\}$ $(u\in V(T))$ and $V(T)$. The tournament $T$ is indecomposable if all its modules are trivial; otherwise it is decomposable. The decomposability index of $T$, denoted by $δ(T)$, is the smallest number of arcs of $T$ that must be reversed to make $T$ indecomposable. The first author conjectured that for $n \geq 5$, we have $δ(n) = \left\lceil \frac{n+1}{4} \right\rceil$, where $…
We say that a digraph $D$ is competitive if any pair of vertices has a common out-neighbor in $D$ and that a graph $G$ is competitively orientable if there exists a competitive orientation of $G$. The notion of competitive digraphs arose while studying digraph whose competition graphs are complete. We derive some useful properties of competitively orientable graphs and show that a complete graph of order $n$ is competitively orientable if and only if $n \geq 7$. Then we completely characterize a competitively orientable complete multipartite graph in terms of the sizes of its partite sets. Mor…
A coloring of a connected graph $G$ is a function $f$ mapping the vertex set of $G$ into the set of all integers. For any subgraph $H$ of $G$, we denote the sum of the values of $f$ on the vertices of $H$ as $f(H)$. If for any integer $k\in \{1,2,\cdots,f(G)\}$, there exists an induced connected subgraph $H$ of $G$ such that $f(H) = k$, then the coloring $f$ is called an IC-coloring of $G$. The IC-index of $G$, denoted as $M(G)$, is the maximum value of $f(G)$ over all possible IC-colorings $f$ of $G$. In this paper, we present a useful method from which a lower bound on the IC-index of any co…
Forward Arc Maximization for Hamilton Oriented Cycles and Paths in Generalizations of Tournaments
Center for Combinatorics and LPMC, Nankai University（南开大学组合数学中心与LPMC）；School of Mathematical Sciences and LPMC, Nankai University（南开大学数学科学学院与LPMC）；School of Science, Hebei University of Technology（河北工业大学理学院）；Department of Mathematics and Computer Science, University of Southern Denmark（南丹麦大学数学与计算机系）；Department of Mathematics, University of Johannesburg（约翰内斯堡大学数学系）；Shenzhen Institutes of Advanced Technology（深圳先进技术研究院）
In 1995, Stiebitz asked the following question: For any positive integers $s,t$, is there a finite integer $f(s,t)$ such that every digraph $D$ with minimum out-degree at least $f(s,t)$ admits a bipartition $(A, B)$ such that $A$ induces a subdigraph with minimum out-degree at least $s$ and $B$ induces a subdigraph with minimum out-degree at least $t$? We give an affirmative answer for tournaments, multipartite tournaments, and digraphs with bounded maximum in-degrees. In particular, we show that for every $ε$ with $0<ε<1/2$, there exists an integer $δ_0$ such that every tournament with minimu…
Let $H$ be a digraph possibly with loops, $D$ a digraph without loops, and $ρ: A(D) \rightarrow V(H)$ a coloring of $A(D)$ ($D$ is said to be an $H$-colored digraph). If $W=(x_{0}, \ldots , x_{n})$ is a walk in $D$, and $i \in \{ 0, \ldots , n-1 \}$, we say that there is an obstruction on $x_{i}$ whenever $(ρ(x_{i-1}, x_{i}), ρ(x_{i}, x_{i+1})) \notin A(H)$ (when $x_{0} = x_{n}$ the indices are taken modulo $n$). We denote by $O_{H}(W)$ the set $\{ i \in \{0, \ldots , n-1 \} :$ there is an obstruction on $x_{i} \}$. The $H$-length of $W$, denoted by $l_{H}(W)$, is defined by $|O_{H}(W)|+1$ whe…
Topological indices are parameters associated with graphs that have many applications in different areas such as mathematical chemistry. Among various topological indices, the Wiener index is classical \cite{w}. In this paper, we prove a formula for the Wiener index and more general $m$-Steiner Wiener index of an arbitrary graph $G$ in terms of the cardinalities of its twin classes. In particular, we will show that calculating these parameters for the graph $G$ can be reduced to calculating the same for a much smaller graph (in general) called the reduced graph of $G$. As applications of our m…
