Related papers: Coloring 3-Colorable Graphs with Low Threshold Ran…

A graph $G$ is 3-colorable if and only if it maps homomorphically to the complete 3-vertex graph $K_3$. The last condition can be checked by a $k$-consistency algorithm where the parameter $k$ has to be chosen large enough, dependent on…

Grotzsch proved that every triangle-free planar graph is 3-colorable. Thomassen proved that every planar graph of girth at least five is 3-choosable. As for other surfaces, Thomassen proved that there are only finitely many 4-critical…

Brooks' theorem states that all connected graphs but odd cycles and cliques can be colored with $\Delta$ colors, where $\Delta$ is the maximum degree of the graph. Such colorings have been shown to admit non-trivial distributed algorithms…

The $q$-Coloring problem asks whether the vertices of a graph can be properly colored with $q$ colors. Lokshtanov et al. [SODA 2011] showed that $q$-Coloring on graphs with a feedback vertex set of size $k$ cannot be solved in time…

This paper is concerned with efficiently coloring sparse graphs in the distributed setting with as few colors as possible. According to the celebrated Four Color Theorem, planar graphs can be colored with at most 4 colors, and the proof…

We study a Tur\'an-type problem on edge-colored complete graphs. We show that for any $r$ and $t$, any sufficiently large $r$-edge-colored complete graph on $n$ vertices with $\Omega(n^{2-1/tr^r})$ edges in each color contains a member from…

Let $G$ be an $n$-vertex graph and let $L:V(G)\rightarrow P(\{1,2,3\})$ be a list assignment over the vertices of $G$, where each vertex with list of size 3 and of degree at most 5 has at least three neighbors with lists of size 2. We can…

An $r$-uniform hypergraph is uniquely $k$-colorable if there exists exactly one partition of its vertex set into $k$ parts such that every edge contains at most one vertex from each part. For integers $k \ge r \ge 2$, let $\Phi_{k,r}$…

The $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for a fixed integer $k$ such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a…

We study Markov chains for randomly sampling $k$-colorings of a graph with maximum degree $\Delta$. Our main result is a polynomial upper bound on the mixing time of the single-site update chain known as the Glauber dynamics for planar…

A linearly ordered (LO) $k$-colouring of a hypergraph is a colouring of its vertices with colours $1, \dots, k$ such that each edge contains a unique maximal colour. Deciding whether an input hypergraph admits LO $k$-colouring with a fixed…

For $k\geq 1$, a $k$-colouring $c$ of $G$ is a mapping from $V(G)$ to $\{1,2,\ldots,k\}$ such that $c(u)\neq c(v)$ for any two non-adjacent vertices $u$ and $v$. The $k$-Colouring problem is to decide if a graph $G$ has a $k$-colouring. For…

Given a large social or information network, how can we partition the vertices into sets (i.e., colors) such that no two vertices linked by an edge are in the same set while minimizing the number of sets used. Despite the obvious practical…

Graph colorings is a fundamental topic in graph theory that require an assignment of labels (or colors) to vertices or edges subject to various constraints. We focus on the harmonious coloring of a graph, which is a proper vertex coloring…

In 1970 Hajnal and Szemer\'edi proved a conjecture of Erd\"os that for a graph with maximum degree $\Delta$, there exists an equitable $\Delta+1$ coloring; that is a coloring where color class sizes differ by at most $1$. In 2007 Kierstand…

A graph G is (d_1,..,d_l)-colorable if the vertex set of G can be partitioned into subsets V_1,..,V_l such that the graph G[V_i] induced by the vertices of V_i has maximum degree at most d_i for all 1 <= i <= l. In this paper, we focus on…

We show that for c >= 2.4682, a random graph on n vertices with c n (1+o(1)) edges almost surely has no 3-colouring. This improves on the current best upper bound of 2.4947.

Given a graph $G$ and color set $\{1, \ldots, k\}$, a $\textit{proper coloring}$ is an assignment of a color to each vertex of $G$ such that no two vertices connected by an edge are given the same color. The problem of drawing a proper…

Vizing's celebrated theorem states that every simple graph with maximum degree $\Delta$ admits a $(\Delta+1)$ edge coloring which can be found in $O(m \cdot n)$ time on $n$-vertex $m$-edge graphs. This is just one color more than the…

This paper studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within $n^{1-\epsilon}$ for any $\epsilon > 0$, is NP-hard in static…