Related papers: Deterministic Simple $(\Delta+\varepsilon\alpha)$-…
In spite of the extensive studies of the 3-coloring problem with respect to several basic parameters, the complexity status of the 3-coloring problem on graphs with small diameter, i.e. with diameter 2 or 3, has been a longstanding and…
In the vertex connectivity problem, given an undirected $n$-vertex $m$-edge graph $G$, we need to compute the minimum number of vertices that can disconnect $G$ after removing them. This problem is one of the most well-studied graph…
We give an online algorithm that with high probability computes a $\left(\frac{e}{e-1} + o(1)\right)\Delta$ edge coloring on a graph $G$ with maximum degree $\Delta = \omega(\log n)$ under online edge arrivals against oblivious adversaries,…
The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore…
The palette sparsification theorem (PST) of Assadi, Chen, and Khanna (SODA 2019) states that in every graph $G$ with maximum degree $\Delta$, sampling a list of $O(\log{n})$ colors from $\{1,\ldots,\Delta+1\}$ for every vertex independently…
Let $X$ be a (repetitive) infinite connected simple graph with a finite upper bound $\Delta$ on the vertex degrees. The main theorem states that $X$ admits a (repetitive) limit aperiodic vertex coloring by $\Delta$ colors. This refines a…
The minimum degree algorithm is one of the most widely-used heuristics for reducing the cost of solving large sparse systems of linear equations. It has been studied for nearly half a century and has a rich history of bridging techniques…
A well-known theorem of Vizing states that if $G$ is a simple graph with maximum degree $\Delta$, then the chromatic index $\chi'(G)$ of $G$ is $\Delta$ or $\Delta+1$. A graph $G$ is class 1 if $\chi'(G)=\Delta$, and class 2 if…
In this paper, we revisit the online recoloring problem introduced recently by Azar et al. In online recoloring, there is a fixed set $V$ of $n$ vertices and an initial coloring $c_0: V\rightarrow [k]$ for some $k\in \mathbb{Z}^{>0}$. Under…
In this paper, we introduce a variant of spectral sparsification, called probabilistic $(\varepsilon,\delta)$-spectral sparsification. Roughly speaking, it preserves the cut value of any cut $(S,S^{c})$ with an $1\pm\varepsilon$…
Recently, \citeauthor*{akbari2021locality}~(ICALP 2023) studied the locality of graph problems in distributed, sequential, dynamic, and online settings from a {unified} point of view. They designed a novel $O(\log n)$-locality deterministic…
The Max-Cut problem is known to be NP-hard on general graphs, while it can be solved in polynomial time on planar graphs. In this paper, we present a fixed-parameter tractable algorithm for the problem on `almost' planar graphs: Given an…
We describe a simple deterministic $O( \varepsilon^{-1} \log \Delta)$ round distributed algorithm for $(2\alpha+1)(1 + \varepsilon)$ approximation of minimum weighted dominating set on graphs with arboricity at most $\alpha$. Here $\Delta$…
Given an undirected graph $G$ and integers $c$ and $k$, the Maximum Edge-Colorable Subgraph problem asks whether we can delete at most $k$ edges in $G$ to obtain a graph that has a proper edge coloring with at most $c$ colors. We show that…
Approximate random $k$-colouring of a graph $G$ is a well studied problem in computer science and statistical physics. It amounts to constructing a $k$-colouring of $G$ which is distributed close to {\em Gibbs distribution} in polynomial…
The problem of (vertex) $(\Delta+1)$-coloring a graph of maximum degree $\Delta$ has been extremely well-studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently.…
We present a novel algorithm for edge-coloring of multigraphs. The correctness of this algorithm for multigraphs with $\chi' > \Delta +1$ ($\chi'$ is the chromatic edge number and $\Delta$ is the maximum vertex degree) would prove a long…
Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a…
We present a new algorithm for finding large independent sets in $3$-colorable graphs with small $1$-sided threshold rank. Specifically, given an $n$-vertex $3$-colorable graph whose uniform random walk matrix has at most $r$ eigenvalues…
In vertex recoloring, we are given $n$ vertices with their initial coloring, and edges arrive in an online fashion. The algorithm must maintain a valid coloring by recoloring vertices, at a cost. The problem abstracts a scenario of job…