Related papers: $Z_{DP}(n)$ is upperly bounded by $n^2-(n+3)/2$

DP-coloring (also called correspondence coloring) is a generalization of list coloring that has been widely studied in recent years after its introduction by Dvo\v{r}\'{a}k and Postle in 2015. As the analogue of the chromatic polynomial of…

Let $G=(V,E)$ be a graph. A (proper) $k$-edge-coloring is a coloring of the edges of $G$ such that any pair of edges sharing an endpoint receive distinct colors. A classical result of Vizing ensures that any simple graph $G$ admits a…

An $acyclic$ edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The \emph{acyclic chromatic index} of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and…

This paper considers upper bounds on the oriented chromatic number $\chi_o(G)$, of an oriented graph $G$ in terms of its $2$-dipath chromatic number $\chi_2(G)$, degeneracy $d(G)$, and maximum degree $\Delta(G)$. In particular, we show that…

The \textit{$r$-dynamic choosability} of a graph $G$, written ${\rm ch}_r(G)$, is the least $k$ such that whenever each vertex is assigned a list of at least $k$ colors a proper coloring can be chosen from the lists so that every vertex $v$…

The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $dn$ is bounded in terms of $H$ and $d$. A breakthrough result of…

An $(m,n)$-colored mixed graph $G$ is a graph with its arcs having one of the $m$ different colors and edges having one of the $n$ different colors. A homomorphism $f$ of an $(m,n)$-colored mixed graph $G$ to an $(m,n)$-colored mixed graph…

Let $n(k_1, k_2)$ be the least integer $n$ such that there exists a graph on $n$ vertices in which every vertex is contained in both a clique of size $k_1$ and an independent set of size $k_2$. Recently, Feige and Pauzner showed that ${n(k,…

The chromatic threshold $\delta_\chi(H)$ of a graph $H$ is the infimum of $d>0$ such that the chromatic number of every $n$-vertex $H$-free graph with minimum degree at least $d n$ is bounded by a constant depending only on $H$ and $d$.…

The proper connection number $pc(G)$ of a connected graph $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one path in $G$ such that no two…

DP-coloring (or correspondence coloring) is a generalization of list coloring that has been widely studied since its introduction by Dvo\v{r}\'{a}k and Postle in 2015. As the analogue of the chromatic polynomial of a graph $G$, $P(G,m)$,…

For an integer $r \ge 2$ and an order $n \equiv 1, 3 \pmod{6}$, write $\delta_r(n)$ for the minimum, over all $r$-colourings $\chi : \binom{[n]}{3} \to [r]$, of $\max_{\mathcal{S}} \mathrm{disc}(\mathcal{S}, \chi)$, where the maximum is…

We consider the following extension of the concept of adjacent strong edge colourings of graphs without isolated edges. Two distinct vertices which are at distant at most $r$ in a graph are called $r$-adjacent. The least number of colours…

The $\chi$-stability index ${\rm es}_{\chi}(G)$ of a graph $G$ is the minimum number of its edges whose removal results in a graph with the chromatic number smaller than that of $G$. In this paper three open problems from [European J.\…

We prove that for any graph $G$, the total chromatic number of $G$ is at most $\Delta(G)+2\left\lceil \frac{|V(G)|}{\Delta(G)+1} \right\rceil$. This saves one color in comparison with a result of Hind from 1992. In particular, our result…

The distinguishing chromatic number, $\chi_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $\varphi$, of $G$, such that the only automorphism of $G$ that preserves all colors of $\varphi$ is the identity map.…

The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the least integer $k$ for which $D$ has a coloring with $k$ colors such that there is no monochromatic directed cycle in $D$. The digraphs considered here are finite and may have…

For a graph $G$, let $\chi(G)$ ($\omega(G)$) denote its chromatic (clique) number. A $P_2+P_3$ is the graph obtained by taking the disjoint union of a two-vertex path $P_2$ and a three-vertex path $P_3$. A $\bar{P_2+P_3}$ is the complement…

A proper total colouring of a graph $G$ is called harmonious if it has the further property that when replacing each unordered pair of incident vertices and edges with their colours, then no pair of colours appears twice. The smallest…

Let $G$ be a simple graph with order $n$, maximum degree $\D(G)$, minimum degree $\delta(G)$ and chromatic index $\chi'(G)$, respectively. A graph $G$ is called {\em $\D$-critical} if $\chi'(G)=\D(G)+1$ and $\chi'(H)\textless \chi'(G)$ for…