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

TL;DR

The paper addresses how to color nodes of a rooted tree with the fewest colors while ensuring that all ancestor–descendant pairs have distinct colors, motivated by a rail-yard control problem. It formalizes colorable partitions, proves a set of global inequalities that relate the color-count sequence $A=(a_1,a_2,\dots)$ to the tree's height distribution $N=(n_0,n_1,\dots)$ (and, via depth, to $D=(d_0,d_1,\dots)$), and identifies canonical colorings by height and by depth as constructive realizations of the minimum color bound. It also shows that these conditions are necessary (and sometimes sufficient), but provides explicit counterexamples (colorable partitions that do not correspond to any coloring) to illustrate limitations. The results yield a rigorous framework for balancing signaling across colors in the rail-yard analogy and extend to general rooted trees, with implications for design of distributed control schemes and tree-structured coloring problems. All findings are stated with precise combinatorial inequalities expressed in terms of height- and depth-based node counts, enabling both theoretical analysis and practical guidance for related systems.

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 $i$.

Figures (5)

• Figure 1: Left: a binary tree with root $r$ and $n=5$ nodes. Right: a full binary tree with $2n+1=11$ nodes ($n=5$ internal nodes in black and $n+1=6$ leaves in white). These two trees are in bijection by adding or removing white nodes; see text.
• Figure 2: Left: Canonical coloring by depth. Right: Canonical coloring by height with blue, red, green and white for nodes with height 0, 1, 2 and 3.
• Figure 3: The two possible colorings with 3 colors for the perfect binary tree of height 2.
• Figure 4: The smallest rooted tree (left) and binary tree (right) with non-sufficient conditions.
• Figure 5: The two smallest full binary trees with non-sufficient conditions.