Chickens and Dukes

09/28/2021
by   Carleton Imbens, et al.
0

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 regarding Dukes. In particular, we prove the existence of 3-Dukes in all multipartite tournaments, and we conclude by proving that in any multipartite tournament, either there is a 1-Duke, three 2-Dukes, or four 3-Dukes.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
06/30/2020

Dichotomizing k-vertex-critical H-free graphs for H of order four

For k ≥ 3, we prove (i) there is a finite number of k-vertex-critical (P...
research
07/13/2019

Variable degeneracy on toroidal graphs

Let f be a nonnegative integer valued function on the vertex-set of a gr...
research
03/14/2021

Decomposing and colouring some locally semicomplete digraphs

A digraph is semicomplete if any two vertices are connected by at least ...
research
12/16/2021

On d-panconnected tournaments with large semidegrees

We prove the following new results. (a) Let T be a regular tournament ...
research
11/27/2020

A Galois Connection Approach to Wei-Type Duality Theorems

In 1991, Wei proved a duality theorem that established an interesting co...
research
02/26/2019

Learning Vertex Convolutional Networks for Graph Classification

In this paper, we develop a new aligned vertex convolutional network mod...

Please sign up or login with your details

Forgot password? Click here to reset