On zero-sum Ramsey numbers modulo 3

TL;DR

This work advances the systematic study of zero-sum Ramsey numbers modulo $3$, focusing on graphs with $3 \mid e(G)$. It develops a general upper bound framework via $2$-good graphs and restrictive colourings, and introduces automorphism switchable pendants (ASPs) to handle trees more precisely. The authors prove $R(F, \mathbb{Z}_3) \le n+2$ for forests and identify sharpness when vertex degrees satisfy $\deg(v) \equiv 1 \pmod{3}$; they further obtain $R(T, \mathbb{Z}_3) \le n+2$ for many trees and $\le n+1$ or $n$ for trees with ASPs, yielding exact values for several infinite families. The results unify several bounds under the $2$-good and ASP frameworks and point to open problems, including full determination of $R(T, \mathbb{Z}_3)$ for double-stars, contributing toward a broader theory of zero-sum Ramsey numbers modulo small primes.

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.

Figures (6)

• Figure 1: (Case 1) Illustration of a restrictive colouring of $K_7$ such that the removal of the uniquely coloured edges incident to each vertex is a (blue) $K_1 \cup K_6$, where $x$ is the isolated vertex. The tree $T$ obtained by coalescing three $P_3$ paths at a common vertex $z$ has a zero-sum embedding by associating the vertex $z$ of degree $3$ with the vertex $x$ in $K_n$.
• Figure 2: The double-star in $T - v$ with leaf $x$ is an example of an ASP at $v$, namely since the double-star has an automorphism mapping $x$ to another leaf $y$, where $x$ is a neighbour of $v$ in $T$ but $y$ is not. On the other hand, the vertex $u$ has no ASPs.
• Figure 3: (Case 1) Illustration of two distinctly weighted embeddings of $T'$ in $K_n$, by considering a fixed embedding of $T_x$ on $V'$ with $x$ and $\sigma(x)$ identified with $v_1$ and $v_2$ respectively, and then adding either the edge $\{v, v_1\}$ or $\{v, v_2\}$.
• Figure 4: (Case 2) Illustration of two distinctly weighted embeddings of $T'$ in $K_n$, by considering two monochromatic embeddings of $T_{r_x}$ on $V'$, one with $r_x$ identified with $v_1$ and another with $r_x$ identified with $v_2$, and then adding the edges $\{v, v_1\}$ and $\{v, v_2\}$ to each embedding respectively.
• Figure 5: (Case 3) Illustration of two distinctly weighted embeddings of $T'$ in $K_n$, by considering a fixed embedding of $T_x$ on $V'$ with $z$, $x$ and $\sigma(x)$ identified with $u$, $u_1$ and $u_2$ respectively, and then adding either the edge $\{u, u_1\}$ or $\{u, u_2\}$, along with the edge $\{v, u\}$.

Theorems & Definitions (30)

• Conjecture 1: Burr-Erdős Conjecture BurrErdosConj
• Theorem 2: CARO1994205
• Conjecture 5: CaroRoditty
• Conjecture 6: caro2019problem
• Theorem 7
• proof
• Corollary 8
• Lemma 9
• proof
• Corollary 10