1,916 research outputs found
Bounded colorings of multipartite graphs and hypergraphs
Let be an edge-coloring of the complete -vertex graph . The
problem of finding properly colored and rainbow Hamilton cycles in was
initiated in 1976 by Bollob\'as and Erd\H os and has been extensively studied
since then. Recently it was extended to the hypergraph setting by Dudek, Frieze
and Ruci\'nski. We generalize these results, giving sufficient local (resp.
global) restrictions on the colorings which guarantee a properly colored (resp.
rainbow) copy of a given hypergraph .
We also study multipartite analogues of these questions. We give (up to a
constant factor) optimal sufficient conditions for a coloring of the
complete balanced -partite graph to contain a properly colored or rainbow
copy of a given graph with maximum degree . Our bounds exhibit a
surprising transition in the rate of growth, showing that the problem is
fundamentally different in the regimes and Our
main tool is the framework of Lu and Sz\'ekely for the space of random
bijections, which we extend to product spaces
A Self-Linking Invariant of Virtual Knots
In this paper we introduce a new invariant of virtual knots and links that is
non-trivial for infinitely many virtuals, but is trivial on classical knots and
links. The invariant is initially be expressed in terms of a relative of the
bracket polynomial and then extracted from this polynomial in terms of its
exponents, particularly for the case of knots. This analog of the bracket
polynomial will be denoted {K} (with curly brackets) and called the binary
bracket polynomial. The key to the combinatorics of the invariant is an
interpretation of the state sum in terms of 2-colorings of the associated
diagrams. For virtual knots, the new invariant, J(K), is a restriction of the
writhe to the odd crossings of the diagram (A crossing is odd if it links an
odd number of crossings in the Gauss code of the knot. The set of odd crossings
is empty for a classical knot.) For K a virtual knot, J(K) non-zero implies
that K is non-trivial, non-classical and inequivalent to its planar mirror
image. The paper also condsiders generalizations of the two-fold coloring of
the states of the binary bracket to cases of three and more colors.
Relationships with graph coloring and the Four Color Theorem are discussed.Comment: 36 pages, 22 figures, LaTeX documen
On DP-Coloring of Digraphs
DP-coloring is a relatively new coloring concept by Dvo\v{r}\'ak and Postle
and was introduced as an extension of list-colorings of (undirected) graphs. It
transforms the problem of finding a list-coloring of a given graph with a
list-assignment to finding an independent transversal in an auxiliary graph
with vertex set . In this paper, we
extend the definition of DP-colorings to digraphs using the approach from
Neumann-Lara where a coloring of a digraph is a coloring of the vertices such
that the digraph does not contain any monochromatic directed cycle.
Furthermore, we prove a Brooks' type theorem regarding the DP-chromatic number,
which extends various results on the (list-)chromatic number of digraphs.Comment: 23 pages, 6 figure
- …