3,683 research outputs found
Asymmetric coloring games on incomparability graphs
Consider the following game on a graph : Alice and Bob take turns coloring
the vertices of properly from a fixed set of colors; Alice wins when the
entire graph has been colored, while Bob wins when some uncolored vertices have
been left. The game chromatic number of is the minimum number of colors
that allows Alice to win the game. The game Grundy number of is defined
similarly except that the players color the vertices according to the first-fit
rule and they only decide on the order in which it is applied. The -game
chromatic and Grundy numbers are defined likewise except that Alice colors
vertices and Bob colors vertices in each round. We study the behavior of
these parameters for incomparability graphs of posets with bounded width. We
conjecture a complete characterization of the pairs for which the
-game chromatic and Grundy numbers are bounded in terms of the width of
the poset; we prove that it gives a necessary condition and provide some
evidence for its sufficiency. We also show that the game chromatic number is
not bounded in terms of the Grundy number, which answers a question of Havet
and Zhu
Complexity of Grundy coloring and its variants
The Grundy number of a graph is the maximum number of colors used by the
greedy coloring algorithm over all vertex orderings. In this paper, we study
the computational complexity of GRUNDY COLORING, the problem of determining
whether a given graph has Grundy number at least . We also study the
variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper)
and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring
algorithm, the subgraph induced by the colored vertices must be connected).
We show that GRUNDY COLORING can be solved in time and WEAK
GRUNDY COLORING in time on graphs of order . While GRUNDY
COLORING and WEAK GRUNDY COLORING are known to be solvable in time
for graphs of treewidth (where is the number of
colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot
be solved in time . We also describe an
algorithm for WEAK GRUNDY COLORING, which is therefore
\fpt for the parameter . Moreover, under the ETH, we prove that such a
running time is essentially optimal (this lower bound also holds for GRUNDY
COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we
show that this is the case for graphs belonging to a number of standard graph
classes including chordal graphs, claw-free graphs, and graphs excluding a
fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY
COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with
the two other problems, we show that CONNECTED GRUNDY COLORING is
\np-complete already for colors.Comment: 24 pages, 7 figures. This version contains some new results and
improvements. A short paper based on version v2 appeared in COCOON'1
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