On zero-sum Ramsey numbers modulo 3

TL;DR

This paper systematically studies zero-sum Ramsey numbers modulo 3, establishing new bounds and exact values for various classes of graphs, especially forests and trees, advancing understanding in combinatorial zero-sum problems.

Contribution

It provides new bounds and exact values for zero-sum Ramsey numbers modulo 3 for forests and trees, including tight bounds under certain conditions.

Findings

For every forest with n vertices and divisible by 3 edges, the zero-sum Ramsey number is at most n+2.

The bound n+2 is tight when all vertices have degrees congruent to 1 mod 3.

Exact values are determined for infinite families of trees.

Abstract

We start with a systematic study of the zero-sum Ramsey numbers. For a graph $G$ with $0 \ (\!\!\!\!\mod 3)$ edges, the zero-sum Ramsey number is defined as the smallest positive integer $R(G, \mathbb{Z}_3)$ such that for every $n \geq R(G, \mathbb{Z}_3)$ and every edge-colouring $f$ of $K_n$ using $\mathbb{Z}_3$, there is a zero-sum copy of $G$ in $K_n$ coloured by $f$, that is: $\sum_{e \in E(G)} f(e) \equiv 0 \ (\!\!\!\!\mod 3)$. Only sporadic results are known for these Ramsey numbers, and we discover many new ones. In particular we prove that for every forest $F$ on $n$ vertices and with $0 \ (\!\!\!\!\mod 3)$ edges, $R(F, \mathbb{Z}_3) \leq n+2$, and this bound is tight if all the vertices of $F$ have degrees $1 \ (\!\!\!\!\mod 3)$. We also determine exact values of $R(T, \mathbb{Z}_3)$ for infinite families of trees.