Related papers: Choosability of a weighted path and free-choosabil…

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…

For a graph $G$ with a given list assignment $L$ on the vertices, we give an algebraical description of the set of all weights $w$ such that $G$ is $(L,w)$-colorable, called permissible weights. Moreover, for a graph $G$ with a given list…

Let $F$ be a (possibly improper) edge-coloring of a graph $G$; a vertex coloring of $G$ is \emph{adapted to} $F$ if no color appears at the same time on an edge and on its two endpoints. If for some integer $k$, a graph $G$ is such that…

We study choosability with separation which is a constrained version of list coloring of graphs. A (k,d)-list assignment L on a graph G is a function that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair…

A list assignment of a graph $G$ is a function $L$ that assigns a list $L(v)$ of colors to each vertex $v\in V(G)$. An $(L,d)^*$-coloring is a mapping $\pi$ that assigns a color $\pi(v)\in L(v)$ to each vertex $v\in V(G)$ so that at most…

A graph is $\ell$-choosable if, for any choice of lists of $\ell$ colors for each vertex, there is a list coloring, which is a coloring where each vertex receives a color from its list. We study complexity issues of choosability of graphs…

A (k,d)-list assignment L of a graph G is a mapping that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. A graph G is (k,d)-choosable if there exists an…

A list assignment $L$ for a graph $G$ is an $(\ell,k)$-list assignment if $|L(v)|\geq \ell$ for each $v \in V(G)$ and $|L(u) \cap L(v)| \leq k$ for each $uv \in E(G)$. We say $G$ is $(\ell,k)$-choosable if it admits an $L$-colouring for…

A graph $G$ is equitably $k$-choosable if, for any given $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\lceil\frac{|V(G)|}{k}\rceil$ vertices. A graph is equitably $k$-colorable if the vertex set…

This paper proves the following result: Assume $G$ is a triangle free planar graph, $X$ is an independent set of $G$. If $L$ is a list assignment of $G$ such that $\mid L(v)\mid = 4$ for each vertex $v \in V(G)-X$ and $\mid L(v)\mid = 3$…

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms…

An $(L,d)^*$-coloring is a mapping $\phi$ that assigns a color $\phi(v)\in L(v)$ to each vertex $v\in V(G)$ such that at most $d$ neighbors of $v$ receive colore $\phi(v)$. A graph is called $(m,d)^*$-choosable, if $G$ admits an…

A graph where each vertex $v$ has a list $L(v)$ of available colors is $L$-colorable if there is a proper coloring such that the color of $v$ is in $L(v)$ for each $v$. A graph is $k$-choosable if every assignment $L$ of at least $k$ colors…

A cycle is $2$-colored if its edges are properly colored by two distinct colors. A $(d,s)$-edge colorable graph $G$ is a $d$-regular graph that admits a proper $d$-edge coloring in which every edge of $G$ is in at least $s-1$ $2$-colored…

A graph $G=(V,E)$ is called $(k,k')$-choosable if for any total list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a mapping…

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…

A graph $G$ is $(a,b)$-choosable if for any color list of size $a$ associated with each vertices, one can choose a subset of $b$ colors such that adjacent vertices are colored with disjoint color sets. This paper shows an equivalence…

For given graph $H$ and graphical property $P$, the conditional chromatic number $\chi(H,P)$ of $H$, is the smallest number $k$, so that $V(H)$ can be decomposed into sets $V_1,V_2,\ldots, V_k$, in which $H[V_i]$ satisfies the property $P$,…

We study a weighted-set graph coloring problem in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given subset of $s$…

We study two weighted graph coloring problems, in which one assigns $q$ colors to the vertices of a graph such that adjacent vertices have different colors, with a vertex weighting $w$ that either disfavors or favors a given color. We…