Related papers: A note on adaptable choosability and choosability …

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…

Let $\phi(k)$ be the minimum number of vertices in a non-$k$-choosable $k$-chromatic graph. The Ohba conjecture, confirmed by Noel, Reed and Wu, asserts that $\phi(k) \ge 2k+2$. This bound is tight if $k$ is even. If $k$ is odd, then it is…

For integers $k, r > 0$, a conditional $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex $v$ of degree $d(v)$ in $G$ is adjacent to at least $\min\{r, d(v)\}$ differently colored…

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…

Two cycles are {\em adjacent} if they have an edge in common. Suppose that $G$ is a planar graph, for any two adjacent cycles $C_{1}$ and $C_{2}$, we have $|C_{1}| + |C_{2}| \geq 11$, in particular, when $|C_{1}| = 5$, $|C_{2}| \geq 7$. We…

In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erd\H{os}, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of…

A graph is $k$-clique-extendible if there is an ordering of the vertices such that whenever two $k$-sized overlapping cliques $A$ and $B$ have $k-1$ common vertices, and these common vertices appear between the two vertices $a,b\in…

A \textit{distinguishing coloring} of a graph $G$ is a coloring of the vertices so that every nontrivial automorphism of $G$ maps some vertex to a vertex with a different color. The \textit{distinguishing number} of $G$ is the minimum $k$…

This work investigates structural and computational aspects of list-based graph coloring under interval constraints. Building on the framework of analogous and p-analogous problems, we show that classical List Coloring, $\mu$-coloring, and…

A celebrated result of Thomassen states that not only can every planar graph be colored properly with five colors, but no matter how arbitrary palettes of five colors are assigned to vertices, one can choose a color from the corresponding…

Xu and Wu proved that if every $5$-cycle of a planar graph $G$ is not simultaneously adjacent to $3$-cycles and $4$-cycles, then $G$ is $4$-choosable. In this paper, we improve this result as follows. Let $\{i, j, k, l\} = \{3,4,5,6\}.$ For…

We consider the following list coloring with separation problem of graphs: Given a graph $G$ and integers $a,b$, find the largest integer $c$ such that for any list assignment $L$ of $G$ with $|L(v)|\le a$ for any vertex $v$ and $|L(u)\cap…

Recently, Dvo\v{r}\'ak, Norin, and Postle introduced flexibility as an extension of list coloring on graphs [JGT 19']. In this new setting, each vertex $v$ in some subset of $V(G)$ has a request for a certain color $r(v)$ in its list of…

For a hypergraph G and a positive integer s, let \chi_{\ell} (G,s) be the minimum value of l such that G is L-colorable from every list L with |L(v)|=l for each v\in V(G) and |L(u)\cap L(v)|\leq s for all u, v\in e\in E(G). This parameter…

Given a graph $G=(V, E)$ and a list of available colors $L(v)$ for each vertex $v\in V$, where $L(v) \subseteq \{1, 2, \ldots, k\}$, List $k$-Coloring refers to the problem of assigning colors to the vertices of $G$ so that each vertex…

A proper total weighting of a graph $G$ is a mapping $\phi$ which assigns to each vertex and each edge of $G$ a real number as its weight so that for any edge $uv$ of $G$, $\sum_{e \in E(v)}\phi(e)+\phi(v) \ne \sum_{e \in…

Kr\'al' and Sgall (2005) introduced a refinement of list colouring where every colour list must be subset to one predetermined palette of colours. We call this $(k,\ell)$-choosability when the palette is of size at most $\ell$ and the lists…

Deciding whether a planar graph (even of maximum degree $4$) is $3$-colorable is NP-complete. Determining subclasses of planar graphs being $3$-colorable has a long history, but since Gr\"{o}tzsch's result that triangle-free planar graphs…

Let $c$ be a proper edge colouring of a graph $G=(V,E)$ with integers $1,2,\ldots,k$. Then $k\geq \Delta(G)$, while by Vizing's theorem, no more than $k=\Delta(G)+1$ is necessary for constructing such $c$. On the course of investigating…

In this article, we use a unified approach to prove several classes of planar graphs are DP-$3$-colorable, which extend the corresponding results on $3$-choosability.