Monochromatic subgraphs in iterated triangulations

TL;DR

This paper constructs a specific 2-edge-coloring of iterated triangulations that avoids monochromatic cycles of length five or more, answering a question about monochromatic subgraphs in such graphs.

Contribution

It demonstrates the existence of 2-edge-colorings of iterated triangulations avoiding large monochromatic cycles, extending previous results to certain radius two graphs.

Findings

Existence of 2-edge-colorings with no monochromatic $C_k$ for $k extgreater 4$ in $Tr(n)$

Negative answer to a question about monochromatic cycles in iterated triangulations

Characterization of radius two graphs with monochromatic copies in $Tr(n)$

Abstract

For integers $n\ge 0$, an iterated triangulation $Tr(n)$ is defined recursively as follows: $Tr(0)$ is the plane triangulation on three vertices and, for $n\ge 1$, $Tr(n)$ is the plane triangulation obtained from the plane triangulation $Tr(n-1)$ by, for each inner face $F$ of $Tr(n-1)$, adding inside $F$ a new vertex and three edges joining this new vertex to the three vertices incident with $F$. In this paper, we show that there exists a 2-edge-coloring of $Tr(n)$ such that $Tr(n)$ contains no monochromatic copy of the cycle $C_k$ for any $k\ge 5$. As a consequence, the answer to one of two questions asked by Axenovich, Schade, Thomassen and Ueckerdt is negative. We also determine the radius two graphs $H$ for which there exists $n$ such that every 2-edge-coloring of $Tr(n)$ contains a monochromatic copy of $H$, extending a result of the above authors for radius two trees.