Related papers: Arboricity and Acyclic Chromatic Number

An edge coloring of a graph $G$ is to color all the edges in the graph such that adjacent edges receive different colors. It is acyclic if each cycle in the graph receives at least three colors. Fiam{\v{c}}ik (1978) and Alon, Sudakov and…

A graph $G$ is equitably $k$-list arborable if for any $k$-uniform list assignment $L$, there is an equitable $L$-colouring of $G$ whose each colour class induces an acyclic graph. The smallest number $k$ admitting such a coloring is named…

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index $a'(G)$ of $G$ is the smallest integer $k$ such that $G$ has an acyclic edge coloring using $k$…

Recent results show that the structural similarity of graphs can be characterized by counting homomorphisms to them: the Tree Theorem states that the well-known color-refinement algorithm does not distinguish two graphs G and H if and only…

Suppose that $T$ is an acyclic $r$-uniform hypergraph, with $r\ge 2$. We define the ($t$-color) chromatic Ramsey number $\chi(T,t)$ as the smallest $m$ with the following property: if the edges of any $m$-chromatic $r$-uniform hypergraph…

The Goldberg-Seymour Conjecture for $f$-colorings states that the $f$-chromatic index of a loopless multigraph is essentially determined by either a maximum degree or a maximum density parameter. We introduce an oriented version of…

Arboricity is a graph parameter akin to chromatic number, in that it seeks to partition the vertices into the smallest number of sparse subgraphs. Where for the chromatic number we are partitioning the vertices into independent sets, for…

Consider a graph obtained by taking edge disjoint union of $k$ complete bipartite graphs. Alon, Saks and Seymour conjectured that such graph has chromatic number at most $k+1$. This well known conjecture remained open for almost twenty…

Let $f_k(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of~$H$ in a $k$-coloring of the edges of $K_n$, and let $ex(n,H)$ denote the Tur\'an number of $H$. In place of $f_2(n,H)$ we simply write $f(n,H)$.…

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that every cycle is colored with at least three colors. The acyclic chromatic index $\chiup_{a}'(G)$ of a graph $G$ is the least number of colors in an acyclic edge…

In 1975 Erd\H{o}s initiated the study of the following very natural question. What can be said about the chromatic number of unit distance graphs in $\mathbb{R}^2$ that have large girth? Over the years this question and its natural…

The acyclic matching number of a graph $G$ is the largest size of an acyclic matching in $G$, that is, a matching $M$ in $G$ such that the subgraph of $G$ induced by the vertices incident to an edge in $M$ is a forest. We show that the…

Let $c$ be an edge-coloring of the complete $n$-vertex graph $K_n$. The problem of finding properly colored and rainbow Hamilton cycles in $c$ was initiated in 1976 by Bollob\'as and Erd\H os and has been extensively studied since then.…

A linear forest is an acyclic graph whose each connected component is a path; or in other words, it is an acyclic graph whose maximum degree is at most 2. A linear coloring of a graph $G$ is an edge coloring of $G$ such that the edges in…

We prove that the number of Hamiltonian paths on the complement of an acyclic digraph is equal to the number of cycle covers. As an application, we obtain a new expansion of the chromatic symmetric function of incomparability graphs in…

The chromatic threshold of a graph $H$ is the minimum-degree density above which every $H$-free graph has bounded chromatic number. We study a two-color Ramsey analogue: for graphs $H_1$ and $H_2$, we ask for the minimum-degree density…

The degree chromatic polynomial $Pm(G,k)$ of a graph $G$ counts the number of $k$-colorings in which no vertex has $m$ adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree…

We investigate which graphs H have the property that in every graph with bounded clique number and sufficiently large chromatic number, some induced subgraph is isomorphic to a subdivision of H. In an earlier paper, one of us proved that…

Ramsey's Theorem states that a graph $G$ has bounded order if and only if $G$ contains no complete graph $K_n$ or empty graph $E_n$ as its induced subgraph. The Gy\'arf\'as-Sumner conjecture says that a graph $G$ has bounded chromatic…

For a graph $G$ and a tree-decomposition $(T, \mathcal{B})$ of $G$, the chromatic number of $(T, \mathcal{B})$ is the maximum of $\chi(G[B])$, taken over all bags $B \in \mathcal{B}$. The tree-chromatic number of $G$ is the minimum…