Related papers: Measurable Brooks's Theorem for Directed Graphs

We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this…

Gallai's conjecture asserts that every connected graph on $n$ vertices can be decomposed into $\frac{n+1}{2}$ paths. For general graphs (possibly disconnected), it was proved that every graph on $n$ vertices can be decomposed into…

A (directed) linear forest is a (di)graph whose components are (directed) paths. The linear arboricity $la(F)$ of a (di)graph $F$ is the minimum number of (directed) linear forests required to decompose its edges. Akiyama, Exoo, and Harary…

A graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We first show that for every triangle-free planar graph G and a vertex…

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…

Given a graph $G$ and a list assignment $L$ for $G$, the indicated $L$-colouring game on $G$ is played by two players: Ann and Ben. In each round, Ann chooses an uncoloured vertex $v$, and Ben colours $v$ with a colour from $L(v)$ that is…

An incidence of an undirected graph G is a pair $(v,e)$ where $v$ is a vertex of $G$ and $e$ an edge of $G$ incident with $v$. Two incidences $(v,e)$ and $(w,f)$ are adjacent if one of the following holds: (i) $v = w$, (ii) $e = f$ or (iii)…

The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity…

We investigate the list packing number of a graph, the least $k$ such that there are always $k$ disjoint proper list-colourings whenever we have lists all of size $k$ associated to the vertices. We are curious how the behaviour of the list…

Let $D$ be a digraph, let $p \geq 1$ be an integer, and let $f: V(D) \to \mathbb{N}_0^p$ be a vector function with $f=(f_1,f_2,\ldots,f_p)$. We say that $D$ has an $f$-partition if there is a partition $(D_1,D_2,\ldots,D_p)$ into induced…

We construct Borel graphs which settle several questions in descriptive graph combinatorics. These include "Can the Baire measurable chromatic number of a locally finite Borel graph exceed the usual chromatic number by more than one?" and…

A $k$-colouring (not necessarily proper) of vertices of a graph is called {\it acyclic}, if for every pair of distinct colours $i$ and $j$ the subgraph induced by the edges whose endpoints have colours $i$ and $j$ is acyclic. In the paper…

In 1980, Burr conjectured that every directed graph with chromatic number $2k-2$ contains any oriented tree of order $k$ as a subdigraph. Burr showed that chromatic number $(k-1)^2$ suffices, which was improved in 2013 to $\frac{k^2}{2} -…

The classical Andr\'{a}sfai-Erd\H{o}s-S\'{o}s theorem considers the chromatic number of $K_{r + 1}$-free graphs with large minimum degree, and in the case $r = 2$ says that any $n$-vertex triangle-free graph with minimum degree greater than…

Let G be a triangle-free graph with maximum degree \delta(G). We show that the chromatic number \c{hi}(G) is less than 67(1 + o(1))\delta/ log \delta.

It is shown that for any fixed $c \geq 3$ and $r$, the maximum possible chromatic number of a graph on $n$ vertices in which every subgraph of radius at most $r$ is $c$ colorable is $\tilde{\Theta}\left(n ^ {\frac{1}{r+1}} \right)$ (that…

A tree $T$ in an edge-colored graph is a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be a fixed integer with $2\le k\le n$. For a vertex subset $S \subseteq…

A $k$-star colouring of a graph $G$ is a function $f:V(G)\to\{0,1,\dots,k-1\}$ such that $f(u)\neq f(v)$ for every edge $uv$ of $G$, and every bicoloured connected subgraph of $G$ is a star. The star chromatic number of $G$, $\chi_s(G)$, is…

The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the least number $k$ such that the vertex set of $D$ can be partitioned into $k$ parts each of which induces an acyclic subdigraph. Introduced by Neumann-Lara in 1982, this digraph…

We prove that the vertices of every $(r + 1)$-uniform hypergraph with maximum degree $\Delta$ may be coloured with $c(\frac{\Delta}{d + 1})^{1/r}$ colours such that each vertex is in at most $d$ monochromatic edges. This result, which is…