Related papers: The minimum degree threshold for perfect graph pac…

We provide a degree condition on a regular $n$-vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$-vertex $k$-chromatic separable graphs into $G$. In general, this degree…

For graph $G$, a connected graph $H$ of order $n$ is said to be $G$-good if $r(G,H)=(\chi(G)-1)(n-1)+s(G)$, where $\chi(G)$ is the chromatic number of $G$ and $s(G)$ is the minimum size of a color class in a $\chi(G)$-coloring of $G$. Let…

An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the…

The distinguishing chromatic number of a graph $G$, denoted $\chi_D(G)$, is the minimum number of colours in a proper vertex colouring of $G$ that is preserved by the identity automorphism only. Collins and Trenk proved that $\chi_D(G)\le…

The degree set of a finite simple graph $G$ is the set of distinct degrees of vertices of $G$. A theorem of Kapoor, Polimeni & Wall asserts that the least order of a graph with a given degree set $\mathscr D$ is $1+\max \mathscr D$.…

Graphs considered in this paper are finite, undirected and without loops, but with multiple edges. For an integer $t\geq 1$, denote by $\mathcal{MG}_t$ the class of graphs whose maximum multiplicity is at most $t$. A graph $G$ is called…

For a finite group $G$, let $\Delta(G)$ denote the character graph built on the set of degrees of the irreducible complex characters of $G$. In graph theory, a perfect graph is a graph $\Gamma$ in which the chromatic number of every induced…

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…

A graph $G$ is perfectly divisible if, for every induced subgraph $H$ of $G$, either $V(H)$ is a stable set or admits a partition into two sets $X_1$ and $X_2$ such that $\omega(H[X_1]) < \omega(H)$ and $H[X_2]$ is a perfect graph. In this…

Let $G$ be an edge-colored graph with $n$ vertices. A subgraph $H$ of $G$ is called a rainbow subgraph of $G$ if the colors of each pair of the edges in $E(H)$ are distinct. We define the minimum color degree of $G$ to be the smallest…

A 2-hued coloring of a graph $G$ (also known as conditional $(k, 2)$-coloring and dynamic coloring) is a coloring such that for every vertex $v\in V(G)$ of degree at least $2$, the neighbors of $v$ receive at least $2$ colors. The smallest…

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…

A vertex coloring of a graph $G$ is distinguishing if non-identity automorphisms do not preserve it. The distinguishing number, $D(G)$, is the minimum number of colors required for such a coloring and the distinguishing threshold,…

The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $p$ such that vertices of $G$ can be partitioned into disjoint classes $X_{1}, ..., X_{p}$ where vertices in $X_{i}$ have pairwise distance greater than…

We consider infinite graphs. The distinguishing number $D(G)$ of a graph $G$ is the minimum number of colours in a vertex colouring of $G$ that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is…

Let $H$ be a fixed graph on $v$ vertices. For an $n$-vertex graph $G$ with $n$ divisible by $v$, an $H$-{\em factor} of $G$ is a collection of $n/v$ copies of $H$ whose vertex sets partition $V(G)$. In this paper we consider the threshold…

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 $G$ be a connected graph of order $n$. A $\{P_3,P_4,P_5\}$-factor is a spanning subgraph $H$ of $G$ such that every component of $H$ is isomorphic to an element of $\{P_3,P_4,P_5\}$. In this paper, we establish a sufficient condition on…

In this paper, we introduce a new concept in graph coloring, namely the \textit{packing total coloring}, which extends the idea of packing coloring to both the vertices and the edges of a given graph. More precisely, for a graph $G$, a…

Let $G$ be a simple graph, and let $n$, $\Delta(G)$ and $\chi' (G)$ be the order, the maximum degree and the chromatic index of $G$, respectively. We call $G$ overfull if $|E(G)|/\lfloor n/2\rfloor > \Delta(G)$, and critical if $\chi'(H) <…