在线图上使用 $(1+o(1))Δ$ 种颜色采样正常着色

Breaking the $2^n$ barrier for 5-coloring and 6-coloring

The coloring problem (i.e., computing the chromatic number of a graph) can be solved in $O^*(2^n)$ time, as shown by Björklund, Husfeldt and Koivisto in 2009. For $k=3,4$, better algorithms are known for the $k$-coloring problem. $3$-coloring can be solved in $O(1.33^n)$ time (Beigel and Eppstein, 2005) and $4$-coloring can be solved in $O(1.73^n)$ time (Fomin, Gaspers and Saurabh, 2007). Surprisingly, for $k>4$ no improvements over the general $O^*(2^n)$ are known. We show that both $5$-coloring and $6$-coloring can also be solved in $O\left(\left(2-\varepsilon\right)^n\right)$ time for some …

Randomly coloring graphs of bounded treewidth

We consider the problem of sampling a proper $k$-coloring of a graph of maximal degree $Δ$ uniformly at random. We describe a new Markov chain for sampling colorings, and show that it mixes rapidly on graphs of bounded treewidth if $k\geq(1+ε)Δ$, for any $ε>0$.

Deterministic Distributed (Delta + o(Δ))-Edge-Coloring, and Vertex-Coloring of Graphs with Bounded Diversity

We consider coloring problems in the distributed message-passing setting. The previously-known deterministic algorithms for edge-coloring employed at least (2Delta - 1) colors, even though any graph admits an edge-coloring with Delta + 1 colors [V64]. Moreover, the previously-known deterministic algorithms that employed at most O(Delta) colors required superlogarithmic time [B15,BE10,BE11,FHK15]. In the current paper we devise deterministic edge-coloring algorithms that employ only Delta + o(Delta) colors, for a very wide family of graphs. Specifically, as long as the arboricity is a = O(Delta…

Superfast Coloring in CONGEST via Efficient Color Sampling

We present a procedure for efficiently sampling colors in the {\congest} model. It allows nodes whose number of colors exceeds their number of neighbors by a constant fraction to sample up to $Θ(\log n)$ semi-random colors unused by their neighbors in $O(1)$ rounds, even in the distance-2 setting. This yields algorithms with $O(\log^* Δ)$ complexity for different edge-coloring, vertex coloring, and distance-2 coloring problems, matching the best possible. In particular, we obtain an $O(\log^* Δ)$-round CONGEST algorithm for $(1+ε)Δ$-edge coloring when $Δ\ge \log^{1+1/\log^*n} n$, and a poly($\…

一种颜色在半流式计算中部分着色的可处理性差异

One Color Makes All the Difference in the Tractability of Partial Coloring in Semi-Streaming

本文研究了半流式计算中k-部分着色的可处理性，发现仅当颜色数为k+1时，k-部分着色可高效解决，而当颜色数为k时则不可行。

Faster Deterministic Distributed Coloring Through Recursive List Coloring

We provide novel deterministic distributed vertex coloring algorithms. As our main result, we give a deterministic distributed algorithm to compute a $(Δ+1)$-coloring of an $n$-node graph with maximum degree $Δ$ in $2^{O(\sqrt{\logΔ})}\cdot\log n$ rounds. This improves on the best previously known time complexity for a large range of values of $Δ$. For graphs with arboricity $a$, we obtain a deterministic distributed algorithm to compute a $(2+o(1))a$-coloring in time $2^{O(\sqrt{\log a})}\cdot\log^2 n$. Further, for graphs with bounded neighborhood independence, we show that a $(Δ+1)$-colorin…

具有固定颜色类大小的着色采样

Sampling Colorings with Fixed Color Class Sizes

本文提出了一种多项式时间算法，用于近似均匀随机生成具有固定颜色类大小的公平着色，适用于q>2Δ的情况，并扩展到接近公平着色的证明。

Linear-time algorithm for vertex 2-coloring without monochromatic triangles on planar graphs

In the problem of 2-coloring without monochromatic triangles (or triangle-tree 2-coloring), vertices of the simple, connected, undirected graph are colored with either 'black' or 'white' such that there are no 3 mutually adjacent vertices of the same color. In this paper we are positively answering the question posed in our previous work, namely, if there exists an algorithm solving 2-coloring without monochromatic triangles on planar graphs with linear-time complexity.

去中心化分布式图着色 II：虚拟图的 degree+1 着色

Decentralized Distributed Graph Coloring II: degree+1-Coloring Virtual Graphs

本文首次系统研究虚拟图着色，发现恒定拥塞度的虚拟图可近乎普通图速度着色，并给出 $O(\log^4\log n)$ 轮的 deg+1 着色算法。

Hierarchical and Modularly-Minimal Vertex Colorings

Cographs are exactly the hereditarily well-colored graphs, i.e., the graphs for which a greedy vertex coloring of every induced subgraph uses only the minimally necessary number of colors $χ(G)$. We show that greedy colorings are a special case of the more general hierarchical vertex colorings, which recently were introduced in phylogenetic combinatorics. Replacing cotrees by modular decomposition trees generalizes the concept of hierarchical colorings to arbitrary graphs. We show that every graph has a modularly-minimal coloring $σ$ satisfying $|σ(M)|=χ(M)$ for every strong module $M$ of $G$.…

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 $k$. 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 $O^*(2.443^n)$ and WEAK G…

Acyclic and Star Colorings of Cographs

An \emph{acyclic coloring} of a graph is a proper vertex coloring such that the union of any two color classes induces a disjoint collection of trees. The more restricted notion of \emph{star coloring} requires that the union of any two color classes induces a disjoint collection of stars. We prove that every acyclic coloring of a cograph is also a star coloring and give a linear-time algorithm for finding an optimal acyclic and star coloring of a cograph. If the graph is given in the form of a cotree, the algorithm runs in O(n) time. We also show that the acyclic chromatic number, the star ch…

从b-着色到$b^*$-着色：大围长与参数化复杂性

From b-Coloring to $b^*$-Coloring: Large Girth and Parameterized Complexity

该研究从b-着色扩展到$b^*$-着色，证明围长≥7的图是$b^*$-单调的，发现围长≥5的d-正则图$b^*$-色数为d+1，还明确了$b^*$-着色的参数化复杂性与b-着色的关联及特定参数下算法的差异。