Polychromatic colorings on the hypercube

Abstract

Given a subgraph G of the hypercube Q n , a coloring of the edges of Q n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube is called the polychromatic number of G. To determine polychromatic numbers, it is only necessary to consider a specific type of coloring, which we call simple. The main tool for finding upper bounds on polychromatic numbers is to translate the question of polychromatically coloring the hypercube so every embedding of a graph G contains every color into a question of coloring the 2-dimensional grid so that every so-called shape sequence corresponding to G contains every color. After surveying the tools for finding polychromatic numbers, we apply these techniques to find polychromatic numbers of a class of graphs called punctured hypercubes. We also consider the problem of finding polychromatic numbers in the setting where larger subcubes of the hypercube are colored. We exhibit two new constructions which show that this problem is not a straightforward generalization of the edge coloring problem.Recall that for an edge e ∈ E(Q n ), l(e) is the number of 1's to the left of the star in e, and r(e) is the number of 1's to the right. Call a coloring χ of the hypercube simple if χ(e) is determined by l(e) and r(e) (such colorings were called Ramsey in [6]). The following lemma tells us that when studying polychromatic colorings on the hypercube, we need only consider simple ones. The proof is essentially from [6], building on ideas from [1].Lemma 3 Let k ≥ 1 and G be a subgraph of Q k . If p(G) = r, then there is a simple G-polychromatic r-coloring on Q k .

Linear Polychromatic Colorings of Hypercube Faces

“…It is then natural to wonder whether an analog of Offner's result holds for ℓ > 1. In a few small cases it was recently shown this is not the case; Goldwasser, Lidicky, Martin, Offner, Talbot and Young prove in [3] the following result.…”

Maximum density of vertex-induced perfect cycles and paths in the hypercube

“…In [1,17,24] results were obtained on the polychromatic number of Q d in Q n , the maximum number of colors in an edge coloring of a large Q n such that every sub-d-cube gets all colors.…”

Polychromatic colorings of complete graphs with respect to 1‐, 2‐factors and Hamiltonian cycles

“…Goldwasser et al . considered the case where is the family of all subgraphs of isomorphic to a minus an edge or a minus a vertex.…”

“…Offner [11] proved that the lower bound is tight for all sufficiently large values of n. Bialostocki [4] treated the special case when d = 2 and n ≥ 2. Goldwasser et al [9] considered the case where H is the family of all subgraphs of Q n isomorphic to a Q d minus an edge or a Q d minus a vertex.If T is a tree and H is the set of all paths of length at least r, then poly H (T ) = ⌈r/2⌉, as was shown by Bollobás et al [5]. When G = K n and H is the set of all r-vertex cliques, poly H (G) was considered by Erdős and Gyárfás [6,10] with the respective colorings called balanced.…”

“…Offner [11] proved that the lower bound is tight for all sufficiently large values of n. Bialostocki [4] treated the special case when d = 2 and n ≥ 2. Goldwasser et al [9] considered the case where H is the family of all subgraphs of Q n isomorphic to a Q d minus an edge or a Q d minus a vertex.…”