8,479 research outputs found
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
Variations on the Roy-Gallai theorem
Abstract.: A generalization of the Roy-Gallai Theorem on the chromatic number of a graph is derived which is also an extension of several other results of Berge and of Li. A simple inductive proof is given which provides a direct way of deriving the Theorem of Li. We also show that some classical results valid for optimal colorings cannot be transposed to suboptimal colorings. We finally investigate some elementary properties which are also valid in suboptimal coloring
On the rooted Tutte polynomial
The Tutte polynomial is a generalization of the chromatic polynomial of graph
colorings. Here we present an extension called the rooted Tutte polynomial,
which is defined on a graph where one or more vertices are colored with
prescribed colors. We establish a number of results pertaining to the rooted
Tutte polynomial, including a duality relation in the case that all roots
reside around a single face of a planar graph. The connection with the Potts
model is also reviewed.Comment: plain latex, 14 pages, 2 figs., to appear in Annales de l'Institut
Fourier (1999
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