相关论文: $2^{(\log N)^{1/10-o(1)}}$ Hardness for Hypergraph…

We introduce a generalization of the well known graph (vertex) coloring problem, which we call the problem of \emph{component coloring of graphs}. Given a graph, the problem is to color the vertices using minimum number of colors so that…

As an application of Szemeredi's regularity lemma, Erdos-Frankl-Rodl (1986) showed that the number of graphs on vertex set {1,2,...n} with a monotone class P is $2^{(1+o(1))ex(n,P)n^2/2}$ where $ex(n,P)$ is the maximum number of edges of an…

A proper edge coloring of a graph is strong if it creates no bichromatic path of length three. It is well known that for a strong edge coloring of a $k$-regular graph at least $2k-1$ colors are needed. We show that a $k$-regular graph…

A graph is weakly $2$-colored if the nodes are labeled with colors black and white such that each black node is adjacent to at least one white node and vice versa. In this work we study the distributed computational complexity of weak…

Since the seminal result of Karger, Motwani, and Sudan, algorithms for approximate 3-coloring have primarily centered around SDP-based rounding. However, it is likely that important combinatorial or algebraic insights are needed in order to…

An edge labeling of a graph distinguishes neighbors by sets (multisets, resp.), if for any two adjacent vertices $u$ and $v$ the sets (multisets, resp.) of labels appearing on edges incident to $u$ and $v$ are different. In an analogous way…

We construct 2-query, quasi-linear size probabilistically checkable proofs (PCPs) with arbitrarily small constant soundness, improving upon Dinur's 2-query quasi-linear size PCPs with soundness $1-\Omega(1)$. As an immediate corollary, we…

This note concerns the trade-off between the degree of the constraint graph and the gap in hardness of approximating the Min-Rep variant of Label Cover (aka Projection Game). We make a very simple observation that, for NP-hardness with gap…

Cut and spectral sparsification of graphs have numerous applications, including e.g. speeding up algorithms for cuts and Laplacian solvers. These powerful notions have recently been extended to hypergraphs, which are much richer and may…

A well-studied coloring problem is to assign colors to the edges of a graph $G$ so that, for every pair of vertices, all edges of at least one shortest path between them receive different colors. The minimum number of colors necessary in…

We show that for every $k\in\mathbb{N}$ and $\varepsilon>0$, for large enough alphabet $R$, given a $k$-CSP with alphabet size $R$, it is NP-hard to distinguish between the case that there is an assignment satisfying at least…

We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time…

A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and the sums of labels taken over the distinct hyperedges are distinct. Let s(H) be the smallest integer N such that there is a sum-distinguishing labeling…

The orthogonality dimension of a graph $G$ over $\mathbb{R}$ is the smallest integer $k$ for which one can assign a nonzero $k$-dimensional real vector to each vertex of $G$, such that every two adjacent vertices receive orthogonal vectors.…

We obtain the following new coloring results: * A 3-colorable graph on $n$ vertices with maximum degree~$\Delta$ can be colored, in polynomial time, using $O((\Delta \log\Delta)^{1/3} \cdot\log{n})$ colors. This slightly improves an…

A {\em packing coloring} of a graph $G$ is a mapping assigning a positive integer (a color) to every vertex of $G$ such that every two vertices of color $k$ are at distance at least $k+1$. The least number of colors needed for a packing…

We give a new randomized distributed algorithm for the $\Delta+1$-list coloring problem. The algorithm and its analysis dramatically simplify the previous best result known of Chang, Li, and Pettie [SICOMP 2020]. This allows for numerous…

A partition $(C_1,C_2,...,C_q)$ of $G = (V,E)$ into clusters of strong (respectively, weak) diameter $d$, such that the supergraph obtained by contracting each $C_i$ is $\ell$-colorable is called a strong (resp., weak) $(d,…

We investigate the computational complexity of the empire colouring problem (as defined by Percy Heawood in 1890) for maps containing empires formed by exactly $r > 1$ countries each. We prove that the problem can be solved in polynomial…

One of the most interesting new developments in hypergraph colourings in these last few years has been Voloshin's notion of colourings of mixed hypergraphs. In this paper we shall study a specific instance of Voloshin's idea: a…