The Graph of Critical Pairs of a Crown

TL;DR

This paper investigates the relationship between the chromatic number and the dimension of a special class of posets called crowns, establishing that for these, the two parameters are equal and providing detailed analysis of their critical pair graphs.

Contribution

The paper proves that for crown posets, the chromatic number of the critical pair graph equals the poset's dimension, and characterizes the independence number and maximal independent sets.

Findings

Chromatic number of G_n^k equals the dimension of S_n^k.

Independence number of G_n^k is (k+1)(k+2)/2.

Detailed analysis of maximal independent sets in G_n^k.

Abstract

There is a natural way to associate with a poset $P$ a hypergraph $H$, called the hypergraph of critical pairs, so that the dimension of $P$ is exactly equal to the chromatic number of $H$. The edges of $H$ have variable sizes, but it is of interest to consider the graph $G$ formed by the edges of $H$ that have size~2. The chromatic number of $G$ is less than or equal to the dimension of $P$ and the difference between the two values can be arbitrarily large. Nevertheless, there are important instances where the two parameters are the same, and we study one of these in this paper. Our focus is on a family $\{S_n^k:n\ge 3, k\ge 0\}$ of height two posets called crowns. We show that the chromatic number of the graph $G_n^k$ of critical pairs of the crown $S_n^k$ is the same as the dimension of $S_n^k$, which is known to be $\lceil 2(n+k)/(k+2)\rceil$. In fact, this theorem follows as an immediate corollary to the stronger result: The independence number of $G_n^k$ is $(k+1)(k+2)/2$. We obtain this theorem as part of a comprehensive analysis of independent sets in $G_n^k$ including the determination of the second largest size among the maximal independent sets, both the reversible and non-reversible types.