Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2311.11005

Rainbow and Gallai-Rado numbers involving binary function equations

Xueliang Li·Yuan Si
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $\mathcal{E}$, $\mathcal{E}_1$, and $\mathcal{E}_2$ be equations, $n$ and $k$ be positive integers. The rainbow number $\operatorname{rb}([n],\mathcal{E})$ is difined as the minimum number of colors such that for every exact $(\operatorname{rb}([n],\mathcal{E}))$-coloring of $[n]$, there exists a rainbow solution of $\mathcal{E}$. The Gallai-Rado number $\operatorname{GR}_k(\mathcal{E}_1:\mathcal{E}_2)$ is defined as the minimum positive integer $N$, if it exists, such that for all $n\ge N$, every $k$-colored $[n]$ contains either a rainbow solution of $\mathcal{E}_1$ or a monochromatic solution of $\mathcal{E}_2$. In this paper, we get some exact values of rainbow and Gallai-Rado numbers involving binary function equations. We also provide an algorithm to calculate the rainbow numbers of nonlinear binary function equations.

Explore related subjects

math.COmath.NT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xueliang Li, Yuan Si. 2024-05-24. Rainbow and Gallai-Rado numbers involving binary function equations. https://arxiv.org/abs/2311.11005

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adjunctions, Box Products, and Forcing Families

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. For bipartite graphs containing a cycle, the forcing conjecture further asserts that asymptotic equality characterizes quasi-random graphs. We establish an adjoint identity for a general class of graph-substitution operators and use it to obtain Sidorenko and forcing results for balanced blow-ups, subdivisions, Cartesian products, and strong products.

math.CO↗
arXiv2026

On the Cost Number of Graphs with Determining Number Two

A distinguishing vertex coloring of a graph $G$ is a vertex coloring such that only the identity automorphism of $G$ preserves the coloring. A graph is $2$-distinguishable if it admits a distinguishing vertex coloring with two colors, and its cost $ρ(G)$ is the minimum size of a color class in such a coloring. The determining number of a graph $G$, denoted by $Det(G)$, is the minimum size of a subset $S\subseteq V(G)$ such that only the trivial automorphism fixes every element of $S$ pointwise. Boutin (J. Combin. Math. Combin. Comput. 85: 161-171, 2013) asked if $ρ(G)$ and $Det(G)$ can be arbitrarily far apart. While the case for $Det(G) = 1$ is trivial, the answer remained unknown for $Det(G) \ge 2$. In this manuscript, we show that if $Det(G)=2$ then not only is $ρ(G)$ bounded, but in fact $ρ(G) \leq 4$. This is the first resolution of Boutin's question for any nontrivial fixed determining number. Moreover, for every fixed $Det(G)= n$, we construct examples giving a lower bound on any possible upper bound for $ρ(G)$ in terms of $n$.

math.CO↗