A note on totally-omnitonal graphs

TL;DR

This paper characterizes totally-omnitonal graphs, showing that only star forests possess the property that all edge colourings in large complete graphs are represented.

Contribution

It provides a complete classification of totally-omnitonal graphs, identifying them as only star forests, which was previously unknown.

Findings

Only star forests are totally-omnitonal graphs.

Totally-omnitonal graphs are characterized as star forests.

The paper establishes the uniqueness of star forests in this property.

Abstract

Let the edges of the complete graph $K_n$ be coloured red or blue, and let $G$ be a graph with $|V(G)| < n$. Then ot(n,G) is defined to be the minimum integer, if it exists, such that any such colouring of $K_n$ contains a copy of $G$ with $r$ red edges and $b$ blue edges for any $r,b \geq 0$ with $r+b= e(G)$. If ot(n,G) exists for every sufficiently large $n$, we say that $G$ is \emph{omnitonal}. Omnitonal graphs were introduced by Caro, Hansberg and Montejano [arXiv:1810.12375,2019]. Now let $G_1$, $G_2$ be two copies of $G$ with their edges coloured red or blue. If there is a colour-preserving isomorphism from $G_1$ to $G_2$ we say that the 2-colourings of $G$ are equivalent. Now we define tot(n,G) to be the minimum integer, if it exists, such that any such colouring of $K_n$ contains all non-quivalent colourings of $G$ with $r$ red edges and $b$ blue edges for any $r,b \geq 0$ with $r+b= e(G)$. If tot(n, G) exists for every sufficiently large $n$, we say that G is \emph{totally-omnitotal}. In this note we show that the only totally-omnitonal graphs are stars or star forests namely a forest all of whose components are stars.