Alon, Krech, and Szabó [SIAM J. Discrete Math., 21 (2007), pp. 66-72] called an edge-coloring of a hypercube with p colors such that every subcube of dimension d contains every color a d-polychromatic p-coloring. Denote by p d the maximum number of colors with which it is possible to d-polychromatically color any hypercube. We find the exact value of p d for all values of d.
a b s t r a c tFor graphs H, G a classical problem in extremal graph theory asks what proportion of the edges of H a subgraph may contain without containing a copy of G. We prove some new results in the case where H is a hypercube. We use a supersaturation technique of Erdős and Simonivits to give a characterization of a set of graphs such that asymptotically the answer is the same when G is a member of this set and when G is a hypercube of some fixed dimension. We apply these results to a specific set of subgraphs of the hypercube called Fibonacci cubes. Additionally, we use a coloring argument to prove new asymptotic bounds on this problem for a different set of graphs. Finally we prove a new asymptotic bound for the case where G is the cube of dimension 3.
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 .
If G is a graph and scriptH is a set of subgraphs of G, then an edge‐coloring of G is called scriptH‐polychromatic if every graph from scriptH gets all colors present in G. The scriptH‐polychromatic number of G, denoted poly scriptHfalse(Gfalse), is the largest number of colors such that G has an scriptH‐polychromatic coloring. In this article, poly scriptHfalse(Gfalse) is determined exactly when G is a complete graph and scriptH is the family of all 1‐factors. In addition polyscriptHfalse(Gfalse) is found up to an additive constant term when G is a complete graph and scriptH is the family of all 2‐factors, or the family of all Hamiltonian cycles.
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