45 research outputs found
Polychromatic Colorings on the Hypercube
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 structured class of colorings, 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.Comment: 24 page
Large rainbow matchings in large graphs
A \textit{rainbow subgraph} of an edge-colored graph is a subgraph whose
edges have distinct colors. The \textit{color degree} of a vertex is the
number of different colors on edges incident to . We show that if is
large enough (namely, ), then each -vertex graph with
minimum color degree at least contains a rainbow matching of size at least
Equitable edge colored Steiner triple systems
A k-edge coloring of G is said to be equitable if the number of edges, at any vertex, colored with a certain color differ by at most one from the number of edges colored with a different color at the same vertex. An STS(v) is said to be polychromatic if the edges in each triple are colored with three different colors. In this paper, we show that every STS(v) admits a 3-edge coloring that is both polychromatic for the STS(v) and equitable for the underlying complete graph. Also, we show that, for v 1 or 3 (mod 6), there exists an equitable k-edge coloring of K which does not admit any polychromatic STS(v), for k = 3 and k = v - 2
F-WORM colorings: Results for 2-connected graphs
Given two graphs F and G, an F-WORM coloring of G is an assignment of colors to its vertices in such a way that no F-subgraph of G is monochromatic or rainbow. If G has at least one such coloring, then it is called F-WORM colorable and W−(G,F) denotes the minimum possible number of colors. Here, we consider F-WORM colorings with a fixed 2-connected graph F and prove the following three main results: (1) For every natural number k, there exists a graph G which is F-WORM colorable and W−(G,F)=k; (2) It is NP-complete to decide whether a graph is F-WORM colorable; (3) For each k≥|V(F)|−1, it is NP-complete to decide whether a graph G satisfies W−(G,F)≤k. This remains valid on the class of F-WORM colorable graphs of bounded maximum degree. We also prove that for each n≥3, there exists a graph G and integers r and s such that s≥r+2, G has Kn-WORM colorings with exactly r and also with s colors, but it admits no Kn-WORM colorings with exactly r+1,…,s−1 colors. Moreover, the difference s−r can be arbitrarily large. © 2017 Elsevier B.V
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs