3,784 research outputs found
A characterization of b-chromatic and partial Grundy numbers by induced subgraphs
Gy{\'a}rf{\'a}s et al. and Zaker have proven that the Grundy number of a
graph satisfies if and only if contains an induced
subgraph called a -atom.The family of -atoms has bounded order and
contains a finite number of graphs.In this article, we introduce equivalents of
-atoms for b-coloring and partial Grundy coloring.This concept is used to
prove that determining if and (under
conditions for the b-coloring), for a graph , is in XP with parameter .We
illustrate the utility of the concept of -atoms by giving results on
b-critical vertices and edges, on b-perfect graphs and on graphs of girth at
least
An easy subexponential bound for online chain partitioning
Bosek and Krawczyk exhibited an online algorithm for partitioning an online
poset of width into chains. We improve this to with a simpler and shorter proof by combining the work of Bosek &
Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of
ladder-free posets. We also provide examples illustrating the limits of our
approach.Comment: 23 pages, 11 figure
Grundy Coloring & Friends, Half-Graphs, Bicliques
The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order ?, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering ?, i.e., the maximum number of colors that the first-fit coloring requires over all possible vertex orderings. Since its inception by Grundy in 1939, Grundy Coloring has been examined for its structural and algorithmic aspects. A brute-force f(k)n^{2^{k-1}}-time algorithm for Grundy Coloring on general graphs is not difficult to obtain, where k is the number of colors required by the most adversarial vertex ordering. It was asked several times whether the dependency on k in the exponent of n can be avoided or reduced, and its answer seemed elusive until now. We prove that Grundy Coloring is W[1]-hard and the brute-force algorithm is essentially optimal under the Exponential Time Hypothesis, thus settling this question by the negative.
The key ingredient in our W[1]-hardness proof is to use so-called half-graphs as a building block to transmit a color from one vertex to another. Leveraging the half-graphs, we also prove that b-Chromatic Core is W[1]-hard, whose parameterized complexity was posed as an open question by Panolan et al. [JCSS \u2717]. A natural follow-up question is, how the parameterized complexity changes in the absence of (large) half-graphs. We establish fixed-parameter tractability on K_{t,t}-free graphs for b-Chromatic Core and Partial Grundy Coloring, making a step toward answering this question. The key combinatorial lemma underlying the tractability result might be of independent interest
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