Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1912.11560

(t,r) broadcast domination in the infinite grid

Rebekah Herrman·Peter van Hintum
View original source ↗Read source document ↗Export metadata ↓

Abstract

The $(t,r)$ broadcast domination number of a graph $G$, $γ_{t,r}(G)$, is a generalization of the domination number of a graph. $γ_{t,r}(G)$ is the minimal number of towers needed, placed on vertices of $G$, each transmitting a signal of strength $t$ which decays linearly, such that every vertex receives a total amount of at least $r$ signal. In this paper we prove a conjecture by Drews, Harris, and Randolph about the minimal density of towers in $\mathbb{Z}^2$ that provide a $(t,3)$ domination broadcast for $t>17$ and explore generalizations. Additionally, we determine the $(t,r)$ broadcast domination number of powers of paths, $P_n^{(k)}$ and powers of cycles, $C_n^{(k)}$.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Rebekah Herrman, Peter van Hintum. 2019-12-24. (t,r) broadcast domination in the infinite grid. https://arxiv.org/abs/1912.11560

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

KEEP EXPLORING

Related papers

arXiv2026

Rooted Spider Embeddings and the Erd\H os-Sós Conjecture

Under a local density condition, we prove that every $k$-edge spider embeds at any prescribed center of degree at least $k$, unless all legs are even and the host graph has one of two specified structures. These structures contain complete bipartite subgraphs with prescribed neighborhoods. The proof uses path rerouting and three exchange lemmas that describe equality in neighborhood estimates. As a consequence, we recover the Erd\H os-Sós bound for all spiders.

math.CO↗
arXiv2026

Random algebraic constructions for extremal and Ramsey problems

Building on Bukh's random algebraic method, we develop a framework for extremal and Ramsey problems involving apex hypergraphs. If $\mathcal{H}$ is a $(d-1)$-partite $(d-1)$-uniform hypergraph with $S$ edges and $\mathcal{H}(t)$ is obtained by adjoining $t$ vertices with common link $\mathcal{H}$, we prove that $\operatorname{ex}(n,\mathcal{H}(t))=Ω_{\mathcal{H}}(n^{d-1/S})$ for $t>9^{S+o_d(S)}$, which is best possible when $\mathcal{H}$ is Sidorenko. Our framework also yields sharper sided Zarankiewicz bounds, quantitative generalized Tur'an bounds, and diagonal multicolor Ramsey constructions. For each fixed $s\geq 2$ and $K\geq 3$, we further prove $\operatorname{r}_K(\mathcal K_{s,t};\mathcal K_n) =Θ_{s,t,K}((n/\log n)^s)$ for $t>9^{s+o(s)}$, extending a theorem of Alon and Rödl from factorial to exponential $t$. The main ingredients are interpolation on $m$-independent varieties, control of the dependencies imposed by symmetry, and linear spaces of forms whose nonzero members remain regular after a common algebraic slice. Limited edge independence then gives the spectral and local-density estimates needed for the Ramsey application.

math.CO↗
arXiv2026

A note on vertex-critical induced subgraphs of shift graphs

Shift graphs, introduced by Erdős and Hajnal in 1964, form one of the simplest known non-recursive constructions of triangle-free graphs with arbitrarily large chromatic number. In this note, we identify a surprising property: for each integer $k \geq 1$, the smallest $k$-chromatic shift graph contains a \emph{unique} induced $k$-vertex-critical subgraph. We give an explicit description of this subgraph and prove its uniqueness. This provides a new family of vertex-critical triangle-free graphs of arbitrarily large chromatic number.

math.CO↗