Related papers: On the Packing Coloring Gap of Graphs
Coloring a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…
The smallest integer $k$ needed for the assignment of colors to the elements so that the coloring is proper (vertices and edges) is called the total chromatic number of a graph. Vizing and Behzed conjectured that the total coloring can be…
Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A…
In the vertex colouring game on a graph $G$, Maker and Breaker alternately colour vertices of $G$ from a palette of $k$ colours, with no two adjacent vertices allowed the same colour. Maker seeks to colour the whole graph while Breaker…
The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $V_i$, $i\in \{1,\ldots,k\}$, where each $V_i$ is an $i$-packing. In this paper, we…
The list coloring problem is a variation of the classical vertex coloring problem, extensively studied in recent years, where each vertex has a restricted list of allowed colors, and having some variations as the $(\gamma,\mu)$-coloring,…
The cochromatic number $\zeta(G)$ of a graph $G$ is the minimum number of colours needed for a vertex colouring where every colour class is either an independent set or a clique. Let $\chi(G)$ denote the usual chromatic number. Around 1991…
A $k$-coloring of a graph $G$ is a partition of the set of vertices of $G$ into $k$ independent sets, which are called colors. A $k$-coloring is neighbor-locating if any two vertices belonging to the same color can be distinguished from…
In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper $r$-coloring $\varphi$ of a graph $G$. We investigate the problem of finding a proper $r$-coloring of $G$, which is "the…
The {\em acyclic chromatic number} of a graph is the least number of colors needed to properly color its vertices so that none of its cycles has only two colors. The {\em acyclic chromatic index} is the analogous graph parameter for edge…
The clique chromatic number of a graph is the minimum number of colours needed to colour its vertices so that no inclusion-wise maximal clique which is not an isolated vertex is monochromatic. We show that every graph of maximum degree…
The DP-coloring problem is a generalization of the list-coloring problem in which the goal is to find an independent transversal in a certain topological cover of a graph $G$. In the online DP-coloring problem, the cover of $G$ is revealed…
An edge-locating coloring of a simple connected graph $G$ is a partition of its edge set into matchings such that the vertices of $G$ are distinguished by the distance to the matchings. The minimum number of the matchings of $G$ that admits…
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 packing $k$-coloring is a natural variation on the standard notion of graph $k$-coloring, where vertices are assigned numbers from $\{1, \ldots, k\}$, and any two vertices assigned a common color $c \in \{1, \ldots, k\}$ need to be at a…
The chromatic sum of a graph is the smallest sum of colors among all proper colorings with natural numbers. The strength is the minimum number of colors needed to achieve the chromatic sum. We construct for each positive integer k a tree…
We consider two graph colouring problems in which edges at distance at most $t$ are given distinct colours, for some fixed positive integer $t$. We obtain two upper bounds for the distance-$t$ chromatic index, the least number of colours…
The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In 2016 McDiarmid, Mitsche and Pralat noted that around p \approx n^{-1/2} the clique chromatic number…
The maximum edge colouring problem considers the maximum colour assignment to edges of a graph under the condition that every vertex has at most a fixed number of distinct coloured edges incident on it. If that fixed number is $q$ we call…
A graph coloring has bounded clustering if each monochromatic component has bounded size. This paper studies such a coloring, where the number of colors depends on an excluded complete bipartite subgraph. This is a much weaker assumption…