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On a problem of Caro on $\mathbb{Z}_3$-Ramsey number of forests

José D. Alvarado, Lucas Colucci, Roberto Parente

TL;DR

This paper resolves Caro's question by determining the exact zero-sum Ramsey numbers $R(F,\mathbb{Z}_3)$ for forests on $n$ vertices with $3\mid e(F)$ and no isolated vertices, distinguishing cases by vertex-degree congruence modulo $3$. The authors introduce and leverage a robust framework built on switching structures, CC colorings, and the parameters $\alpha_s(F)$, $\alpha_{C_4}(\chi)$, and $\alpha'_l(F)$ to categorize forests into Type 0, Type 1, and Type 2, enabling tight upper and matching lower bounds. The main result states $R(F,\mathbb{Z}_3)=n+2$ when $F$ is $1\pmod{3}$-regular or a star, $R(F,\mathbb{Z}_3)=n+1$ in canonically mixed degree configurations, and $R(F,\mathbb{Z}_3)=n$ otherwise, thereby completing the Caro program for forests. The work provides a versatile toolkit that could be extended to other groups or broader graph families and highlights several open questions for future study.

Abstract

Let $k$ be a positive integer and let $G$ be a graph. The zero-sum Ramsey number $R(G,\mathbb{Z}_k)$ is the least integer $N$ (if it exists) such that for every edge-coloring $χ\, : \, E(K_N) \, \rightarrow \, \mathbb{Z}_k$ one can find a copy of $G$ in $K_N$ such that $\sum_{e \, \in \, E(G)}{χ(e)} \, = \, 0$. In 2019, Caro made a conjecture about the $\mathbb{Z}_3$-Ramsey number of trees. In this paper, we settle this conjecture, fixing an incorrect case, and extend the result to forests. Namely, we show that \begin{equation*} R(F,\mathbb{Z}_3) = \left\{ \begin{array}{ll} n+2, & \text{if $F$ is $1 (\mathrm{mod}\, 3)$ regular or a star;}\\ n+1, & \text{if $3 \nmid d(v)$ for every $v \in V(F)$ or $F$ has exactly one} \\ \phantom{placeholder} & \text{vertex of degree $0 (\mathrm{mod}\, 3)$ and all others are $1 (\mathrm{mod}\, 3)$,} \\ \phantom{placeholder} & \text{and $F$ is not $1 (\mathrm{mod}\, 3)$ regular or a star;}\\ n, & \text{otherwise.} \end{array} \right. \end{equation*} where $F$ is any forest on $n$ vertices with $3\mid e(F)$ and no isolated vertices.

On a problem of Caro on $\mathbb{Z}_3$-Ramsey number of forests

TL;DR

This paper resolves Caro's question by determining the exact zero-sum Ramsey numbers for forests on vertices with and no isolated vertices, distinguishing cases by vertex-degree congruence modulo . The authors introduce and leverage a robust framework built on switching structures, CC colorings, and the parameters , , and to categorize forests into Type 0, Type 1, and Type 2, enabling tight upper and matching lower bounds. The main result states when is -regular or a star, in canonically mixed degree configurations, and otherwise, thereby completing the Caro program for forests. The work provides a versatile toolkit that could be extended to other groups or broader graph families and highlights several open questions for future study.

Abstract

Let be a positive integer and let be a graph. The zero-sum Ramsey number is the least integer (if it exists) such that for every edge-coloring one can find a copy of in such that . In 2019, Caro made a conjecture about the -Ramsey number of trees. In this paper, we settle this conjecture, fixing an incorrect case, and extend the result to forests. Namely, we show that \begin{equation*} R(F,\mathbb{Z}_3) = \left\{ \begin{array}{ll} n+2, & \text{if is regular or a star;}\\ n+1, & \text{if for every or has exactly one} \\ \phantom{placeholder} & \text{vertex of degree and all others are ,} \\ \phantom{placeholder} & \text{and is not regular or a star;}\\ n, & \text{otherwise.} \end{array} \right. \end{equation*} where is any forest on vertices with and no isolated vertices.

Paper Structure

This paper contains 16 sections, 26 theorems, 8 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction and main result
  2. Definitions and notation
  3. Lower bounds
  4. Type $1$ forests
  5. Type $2$ forests
  6. Upper bounds
  7. $1 \pmod 3$ regular graphs
  8. A bound for forests with siblings
  9. Forests with $\alpha_s(F) \geq 2$ and colorings with $\alpha_{C_4}(\chi) \geq 2$
  10. Colorings without alternating $C_4$
  11. Colorings with $\alpha_{C_4}(\chi) = 1$
  12. Forests with $\alpha_s(F)=1$
  13. One connected component
  14. Two connected components
  15. Three connected components
  16. ...and 1 more sections

Key Result

Theorem 2

Let $F$ be a forest on $n$ vertices with $3\mid e(F)$ and without isolated vertices. Then,

Figures (4)

  • Figure 1: Switching structures $(v_1,v_2,v_3,v_4)$ and $(v_5,v_6,v_7,v_8)$ in a forest
  • Figure 2: A CC$(2,3,3)$ coloring
  • Figure 3: A $B(3,2,2)$, a $C(2,2,1,1,3)$, and a $T(3,1,1,1,3,1)$
  • Figure 4: The four types of generalized switchings

Theorems & Definitions (60)

  • Conjecture 1
  • Theorem 2
  • Remark 3
  • Definition 4: Switching structure
  • Definition 5: Alternating $C_4$
  • Definition 6: CC colorings
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 50 more