Grundy着色和朋友，半图，Bicliques
Grundy Coloring & friends, Half-Graphs, Bicliques
摘要：
首次拟合着色是一种启发式方法，它为每个顶点分配最小的可用颜色，以指定的顺序$σ$到达。Grundy Coloring问题询问对于最敌对的顶点排序$σ$需要多少颜色，即，在所有可能的顶点排序中，第一拟合着色所需的最大颜色数。自1939年Grundy成立以来，Grundy着色一直在研究其结构和算法方面。一般图的Grundy着色的一个强力$f(k)n^{2^{k-1}}$时间算法并不难得到，其中$k$是最具对抗性的顶点排序所需的颜色数。有人问过几次是否可以避免或减少$n$的指数中对$k$的依赖性，直到现在，它的答案似乎难以捉摸。在指数时间假设下，证明了Grundy染色是W[1]-困难的，而蛮力算法是本质最优的，从而从反面解决了这个问题。在我们的W[1]-硬度证明中的关键成分是使用所谓的半图作为构建块来将颜色从一个顶点传递到另一个顶点。利用半图，我们还证明了b-色核是W[1]-硬的，其参数化复杂性是Panolan等人提出的一个开放问题。一个自然的后续问题是，在没有（大的）半图的情况下，参数化的复杂性如何变化。我们建立了固定参数的易处理$K_{t,t}$ -自由图的b-色核和部分Grundy染色，使一个步骤来回答这个问题。易处理性结果背后的关键组合引理可能是独立的利益。
英文摘要：
The first-fit coloring is a heuristic that assigns to each vertex, arriving in a specified order $σ$, the smallest available color. The problem Grundy Coloring asks how many colors are needed for the most adversarial vertex ordering $σ$, 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 '17]. 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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