# Adaptive majority problems for restricted query graphs and for weighted sets

## Main Article Content

Gábor Damásdi Dániel Gerbner Gyula O. H. Katona Abhishek Methuku Balázs Keszegh Dániel Lenger Dániel T. Nagy Dömötör Pálvölgyi Balázs Patkós Máté Vizer Gábor Wiener

## Abstract

Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the \emph{majority color} (if it exists), and any vertex of this color is called a \emph{majority vertex}. We study the problem of finding a majority vertex (or show that none exists), if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by $m(G)$. It was shown by Saks and Werman that $m(K_n)\!=\! n-b(n)$ where $b(n)$ is the number of 1's in the binary representation of $n$. In this paper we initiate the study of the problem for general graphs. The obvious bounds for a connected graph $G$ on $n$ vertices are $n-b(n)\le m(G)\le n-1$. We show that for any tree $T$ on an even number of vertices we have $m(T)=n-1$, and that for any tree $T$ on an odd number of vertices, we have $n-65\le m(T)\le n-2$. Our proof uses results about the weighted version of the problem for $K_n$, which may be of independent interest. We also exhibit a sequence $G_n$ of graphs with $m(G_n)=n-b(n)$ such that the number of edges in $G_n$ is $O(nb(n))$.

## Article Details

How to Cite
Damásdi, G., Gerbner, D., Katona, G., Methuku, A., Keszegh, B., Lenger, D., Nagy, D., Pálvölgyi, D., Patkós, B., Vizer, M., & Wiener, G. (2019). Adaptive majority problems for restricted query graphs and for weighted sets. Acta Mathematica Universitatis Comenianae, 88(3), 601-609. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1272/700
Section
EUROCOMB 2019