Deploying robots with two sensors in $K_{1,6}$-free graphs

TL;DR

This paper proves that certain $K_{1,6}$-free graphs with minimum degree at least two can be labeled with two-element subsets from a five-element set to ensure coverage of each label in each vertex or its neighbors, extending previous results.

Contribution

It generalizes a robotics-related labeling result from 3-regular graphs to a broader class of $K_{1,6}$-free graphs with minimum degree at least two.

Findings

Fractional domatic number at least 5/2

Valid labeling exists for all but eight exceptional graphs

Extends previous 3-regular graph results to $K_{1,6}$-free graphs

Abstract

Let $G$ be a graph of minimum degree at least two with no induced subgraph isomorphic to $K_{1,6}$. We prove that if $G$ is not isomorphic to one of eight exceptional graphs, then it is possible to assign two-element subsets of $\{1,2,3,4,5\}$ to the vertices of $G$ in such a way that for every $i\in\{1,2,3,4,5\}$ and every vertex $v\in V(G)$ the label $i$ is assigned to $v$ or one of its neighbors. It follows that $G$ has fractional domatic number at least $5/2$. This is motivated by a problem in robotics and generalizes a result of Fujita, Yamashita and Kameda who proved that the same conclusion holds for all $3$-regular graphs.