Colorings of (r, r)-Uniform, Complete, Circular, Mixed Hypergraphs

Abstract

In colorings of some block designs, the vertices of blocks can be thought of as hyperedges of a hypergraph H that can be placed on a circle and colored according to some rules that are related to colorings of circular mixed hypergraphs. A mixed hypergraph H is called circular if there exists a host cycle on the vertex set X such that every edge (C- or D-) induces a connected subgraph of this cycle. We propose an algorithm to color the (r,r)-uniform, complete, circular, mixed hypergraphs for all feasible values with no gaps. In doing so, we show χ(H)=2 and χ¯(H)=n−s or n−s−1 where s is the sieve number.

Minimal Non-C-Perfect Hypergraphs with Circular Symmetry

“…Other constructions of non-C-perfect hypergraphs and further results on C-perfectness can be found in [6][7][8][9][10].…”

Coloring Properties of Mixed Cycloids

“…The smallest example F 5 is shown in Figure 1. Recent works deal with KC(n; p, p), namely [14] proves χ = n 2 for p = q = 3, and the main theorem of [15] gives n − s − 1 ≤ χ ≤ n − s if p = q > 3, where s denotes the "sieve number". We shall give the definition and some comments on this parameter in the concluding section.…”

“…At the end of the paper [15], its authors raise the problem of determining the feasible set of KC(n; p, q) for all p and q. Due to Theorem 1, this reduces to computing the upper chromatic number for all 3-tuples n, p, q.…”

“…Assign pairwise different colors to the remaining vertices of X. The cycloid is now colored with c(H) = n − s or c(H) = n − s − 1 colors, as shown in [15]. Let M denote this number.…”