Remote control system of a binary tree of switches -- I. constraints and inequalities

TL;DR

This paper analyzes a tree coloring model inspired by a remote control system, formalizing constraints on node coloring based on tree structure, and deriving inequalities relating node counts by color, height, and depth.

Contribution

It formalizes coloring constraints in a tree model and establishes inequalities linking node distributions across colors, height, and depth.

Findings

Derived inequalities for node color distributions based on tree height.

Established that coloring constraints apply to nodes by depth as well as height.

Provided a mathematical framework for analyzing switch tree colorings.

Abstract

We study a tree coloring model introduced by Guidon (2018), initially based on an analogy with a remote control system of a rail yard, seen as a switch tree. For a given rooted tree, we formalize the constraints on the coloring, in particular on the minimum number of colors, and on the distribution of the nodes among colors. We show that the sequence $(a_1,a_2,a_3,\cdots)$, where $a_i$ denotes the number of nodes with color $i$, satisfies a set of inequalities which only involve the sequence $(n_0,n_1,n_2,\cdots)$ where $n_i$ denotes the number of nodes with height $i$. By coloring the nodes according to their depth, we deduce that these inequalities also apply to the sequence $(d_0,d_1,d_2,\cdots)$ where $d_i$ denotes the number of nodes with depth $i$.