Related papers: List coloring with requests
We introduce the notion of locally identifying coloring of a graph. A proper vertex-coloring c of a graph G is said to be locally identifying, if for any adjacent vertices u and v with distinct closed neighborhood, the sets of colors that…
A graph $G$ is free $(a,b)$-choosable if for any vertex $v$ with $b$ colors assigned and for any list of colors of size $a$ associated with each vertex $u\ne v$, the coloring can be completed by choosing for $u$ a subset of $b$ colors such…
Coloring the vertices of a graph G subject to given conditions can be considered as a random experiment and corresponding to this experiment, a discrete random variable X can be defined as the colour of a vertex chosen at random, with…
Let $G$ be a plane graph with outer cycle $C$ and let $(L(v):v\in V(G))$ be a family of non-empty sets. By an $L$-coloring of $G$ we mean a (proper) coloring $\phi$ of $G$ such that $\phi(v)\in L(v)$ for every vertex $v$ of $G$. Thomassen…
We completely classify the computational complexity of the list H-colouring problem for graphs (with possible loops) in combinatorial and algebraic terms: for every graph H the problem is either NP-complete, NL-complete, L-complete or is…
List k-Coloring (Li k-Col) is the decision problem asking if a given graph admits a proper coloring compatible with a given list assignment to its vertices with colors in {1,2,..,k}. The problem is known to be NP-hard even for k=3 within…
We investigate Fair and Tolerant (FAT) graph colorings, a coloring framework in which each vertex is allowed to share its color with a prescribed fraction of its neighbors, while the remaining neighbors are required to be distributed evenly…
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…
The \textit{square} of a graph $G$, denoted by $G^2$, is obtained from $G$ by adding an edge to connect every pair of vertices with a common neighbor in $G$. In this paper we prove that for every planar graph $G$ with maximum degree at most…
We study relations between three interrelated notions of graph (list) coloring: single conflict coloring, adapted list coloring and choosability with separation (with $1$ overlapping color between lists of adjacent vertices), and their…
A {\em restraint} on a (finite undirected) graph $G = (V,E)$ is a function $r$ on $V$ such that $r(v)$ is a finite subset of ${\mathbb N}$; a proper vertex colouring $c$ of $G$ is {\em permitted} by $r$ if $c(v) \not\in r(v)$ for all…
Given a graph $G$ and a mapping $f:V(G) \to \mathbb{N}$, an $f$-list assignment of $G$ is a function that maps each $v \in V(G)$ to a set of at least $f(v)$ colors. For an $f$-list assignment $L$ of a graph $G$, a proper conflict-free…
We say that an edge colouring breaks an automorphism if some edge is mapped to an edge of a different colour. We say that the colouring is distinguishing if it breaks every non-identity automorphism. We show that such colouring can be…
Inspired by earlier results about proper and polychromatic coloring of hypergraphs, we investigate such colorings of directed hypergraphs, that is, hypergraphs in which the vertices of each hyperedge is partitioned into two parts, a tail…
Let $G$ be a graph and c a proper k-coloring of G, i.e. any two adjacent vertices u and v have different colors c(u) and c(v). A proper k-coloring is a b-coloring if there exists a vertex in every color class that contains all the colors in…
A graph is $k$-planar if it can be drawn in the plane so that each edge is crossed at most $k$ times. Typically, the class of 1-planar graphs is among the most investigated graph families within the so-called "beyond planar graphs". A…
Given a graph $G$ and a list assignment $L(v)$ for each vertex of $v$ of $G$. A proper $L$-list-coloring of $G$ is a function that maps every vertex to a color in $L(v)$ such that no pair of adjacent vertices have the same color. We say…
We study the problem of approximately counting the number of list packings of a graph. The analogous problem for usual vertex coloring and list coloring has attracted a lot of attention. For list packing the setup is similar but we seek a…
The $k$-Colouring problem is to decide if the vertices of a graph can be coloured with at most $k$ colours for a fixed integer $k$ such that no two adjacent vertices are coloured alike. If each vertex u must be assigned a colour from a…
A majority edge-coloring of a graph without pendant edges is a coloring of its edges such that, for every vertex $v$ and every color $\alpha$, there are at most as many edges incident to $v$ colored with $\alpha$ as with all other colors.…