Related papers: Acyclic subgraphs with high chromatic number

A star edge coloring of a graph is a proper edge coloring with no $2$-colored path or cycle of length four. The star chromatic index $\chi'_{st}(G)$ of $G$ is the minimum number $t$ for which $G$ has a star edge coloring with $t$ colors. We…

Let $\chi(G)$ denote the chromatic number of a graph and $\chi_v(G)$ denote the vector chromatic number. For all graphs $\chi_v(G) \le \chi(G)$ and for some graphs $\chi_v(G) \ll \chi(G)$. Galtman proved that Hoffman's well-known lower…

We discuss the minimal number of vertices in a graph with a large chromatic number such that each ball of a fixed radius in it has a small chromatic number. It is shown that for every graph $G$ on $\sim((n+rc)/(c+rc))^{r+1}$ vertices such…

A cyclic coloring of a plane graph $G$ is a coloring of its vertices such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a plane graph $G$ is its cyclic chromatic number…

In 1985, Mader conjectured the existence of a function $f$ such that every digraph with minimum out-degree at least $f(k)$ contains a subdivision of the transitive tournament of order $k$. This conjecture is still completely open, as the…

A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The \emph{nonrepetitive chromatic number} of a graph $G$ is the…

Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in…

A $k$-cycle in a graph is a cycle of length $k.$ A graph $G$ of order $n$ is called edge-pancyclic if for every integer $k$ with $3\le k\le n,$ every edge of $G$ lies in a $k$-cycle. It seems difficult to determine the minimum size $f(n)$…

The Hall ratio of a graph G is the maximum of |V(H)|/alpha(H) over all subgraphs H of G. Clearly, the Hall ratio of a graph is a lower bound for the fractional chromatic number. It has been asked whether conversely, the fractional chromatic…

For a simple graph $G$, let $\chi_f(G)$ be the fractional chromatic number of $G$. In this paper, we aim to establish upper bounds on $\chi_f(G)$ for those graphs $G$ with restrictions on the clique number. Namely, we prove that for $\Delta…

The packing chromatic number $\chi$ $\rho$ (G) of an undirected (resp. oriented) graph G is the smallest integer k such that its set of vertices V (G) can be partitioned into k disjoint subsets V 1,..., V k, in such a way that every two…

A $k$-majority digraph is a directed graph created by combining $k$ individual rankings on the same ground set to form a consensus where edges point in the direction indicated by a strict majority of the rankings. The $k$-majority digraph…

A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every coloring of the edges of $K_n$ with $r$ colors, there is a cover of its vertex set by at most $f(r) = O(r^2 \log r)$ vertex-disjoint monochromatic cycles. In…

The clique chromatic number of a graph G=(V,E) is the minimum number of colors in a vertex coloring so that no maximal (with respect to containment) clique is monochromatic. We prove that the clique chromatic number of the binomial random…

Given an acyclic oriented graph $\vec{H}$ and a graph $G$, we write $G \to \vec{H}$ if every orientation of $G$ has an oriented copy of $\vec{H}$. We define $\vec{R}(\vec{H})$ as the smallest number $n$ such that there exists a graph $G$…

The \emph{choice number} of a graph $G$, denoted $\ch(G)$, is the minimum integer $k$ such that for any assignment of lists of size $k$ to the vertices of $G$, there is a proper colouring of $G$ such that every vertex is mapped to a colour…

Let $G=(V,E)$ be a graph of density $p$ on $n$ vertices. Following Erd\H{o}s, \L uczak and Spencer, an $m$-vertex subgraph $H$ of $G$ is called {\em full} if $H$ has minimum degree at least $p(m - 1)$. Let $f(G)$ denote the order of a…

For a graph G=(V,E), a hypergraph H is called Berge-G if there is a bijection f from E(G) to E(H) such that for each e in E(G), e is a subset of f(e). The set of all Berge-G hypergraphs is denoted B(G). For integers k>1, r>1, and a graph G,…

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every $2$-coloring of the…

It is well-known that the graphs not containing a given graph H as a subgraph have bounded chromatic number if and only if H is acyclic. Here we consider ordered graphs, i.e., graphs with a linear ordering on their vertex set, and the…