{"title":"Remote control system of a binary tree of switches -- I. constraints and inequalities","authors":"Olivier Golinelli","doi":"arxiv-2405.16938","DOIUrl":null,"url":null,"abstract":"We study a tree coloring model introduced by Guidon (2018), initially based\non an analogy with a remote control system of a rail yard, seen as a switch\ntree. For a given rooted tree, we formalize the constraints on the coloring, in\nparticular on the minimum number of colors, and on the distribution of the\nnodes among colors. We show that the sequence $(a_1,a_2,a_3,\\cdots)$, where\n$a_i$ denotes the number of nodes with color $i$, satisfies a set of\ninequalities which only involve the sequence $(n_0,n_1,n_2,\\cdots)$ where $n_i$\ndenotes the number of nodes with height $i$. By coloring the nodes according to\ntheir depth, we deduce that these inequalities also apply to the sequence\n$(d_0,d_1,d_2,\\cdots)$ where $d_i$ denotes the number of nodes with depth $i$.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"31 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2405.16938","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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 $i$.