Bernoulli trials of fixed parity, random and randomly oriented graphs

TL;DR

The paper introduces a method to bound the median of the number of distinct colors in biased coin coloring, using analysis of parity-restricted Bernoulli trials and random graph orientations.

Contribution

It develops a novel approach linking parity-constrained Bernoulli variables with graph orientation distributions to analyze coloring problems.

Findings

Provides an upper bound on the median of the number of colors.

Establishes a connection between parity-restricted Bernoulli variables and graph properties.

Applies the method to analyze distributions of vertices with even or odd degrees.

Abstract

Suppose you can color $n$ \emph{biased} coins with $n$ colors, all coins having the same bias. It is forbidden to color both sides of a coin with the same color, but all other colors are allowed. Let $X$ be the number of different colors after a toss of the coins. We present a method to obtain an upper bound on a median of $X$. Our method is based on the analysis of the probability distribution of the number of vertices with even in-degree in graphs whose edges are given random orientations. Our analysis applies to the distribution of the number of vertices with odd degree in random sub-graphs of fixed graphs. It turns out that there are parity restrictions on the random variables that are under consideration. Hence, in order to present our result, we introduce a class of Bernoulli random variables whose total number of successes is of fixed parity and are closely related to Poisson trials conditional on the event that their outcomes have fixed parity.