Related papers: Sufficient conditions on cycles that make planar g…

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. If G is a planar graph without pairwise adjacent…

Consider two conditions on a graph: (1) each 5-cycle is not a subgraph of 5-wheel and does not share exactly one edge with 3-cycle, and (2) each 5-cycle is not adjacent to two 3-cycles and is not adjacent to a 4-cycle with chord. We show…

Wang and Lih in 2002 conjectured that every planar graph without adjacent triangles is 4-choosable. In this paper, we prove that every planar graph without any 4-cycle adjacent to two triangles is DP-4-colorable, which improves the results…

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…

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…

DP-coloring is a generalization of a list coloring in simple graphs. Many results in list coloring can be generalized in those of DP-coloring. Kim and Ozeki showed that planar graphs without $k$-cycles where $k=3,4,5,$ or $6$ are…

A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. We prove that if cycles of length at most four in a planar graph G are pairwise far apart, then G is 3-choosable. This is analogous…

By a well-known theorem of Thomassen and a planar graph depicted by Voigt, we know that every planar graph is $5$-choosable, and the bound is tight. In 1999, Lam, Xu and Liu reduced $5$ to $4$ on $C_4$-free planar graphs. In the paper, by…

Let $\mathscr{G}$ be the class of plane graphs without triangles normally adjacent to $8^{-}$-cycles, without $4$-cycles normally adjacent to $6^{-}$-cycles, and without normally adjacent $5$-cycles. In this paper, it is shown that every…

This paper studies the choosability of signed planar graphs. We prove that every signed planar graph is 5-choosable and that there is a signed planar graph which is not 4-choosable while the unsigned graph is 4-choosable. For each $k \in…

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 graph G is (a:b)-colorable if there exists an assignment of b-element subsets of {1,...,a} to vertices of G such that sets assigned to adjacent vertices are disjoint. We show that every planar graph without cycles of length 4 or 5 is…

A graph $G$ is {\em $k$-choosable} if for every assignment of a set $S(v)$ of $k$ colors to every vertex $v$ of $G$, there is a proper coloring of $G$ that assigns to each vertex $v$ a color from $S(v)$. We consider the complexity of…

For positive integers $a$ and $b$, a graph $G$ is $(a:b)$-choosable if, for each assignment of lists of $a$ colors to the vertices of $G,$ each vertex can be colored with a set of $b$ colors from its list so that adjacent vertices are…

Let $G$ be a planar graph with no two 3-cycles sharing an edge. We show that if $\Delta(G)\geq 9$, then $\chi'_l(G) = \Delta(G)$ and $\chi''_l(G)=\Delta(G)+1.$ We also show that if $\Delta(G)\geq 6$, then $\chi'_l(G)\leq\Delta(G)+1$ and if…

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…

Graph coloring with preferences offers a powerful framework for constraint satisfaction problems in which fulfilling every request is impossible but satisfying a guaranteed positive fraction is highly desirable. A \emph{request} on a graph…

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…

The choosability $\chi_\ell(G)$ of a graph $G$ is the minimum $k$ such that having $k$ colors available at each vertex guarantees a proper coloring. Given a toroidal graph $G$, it is known that $\chi_\ell(G)\leq 7$, and $\chi_\ell(G)=7$ if…

A planar graph is essentially $4$-connected if it is 3-connected and every of its 3-separators is the neighborhood of a single vertex. Jackson and Wormald proved that every essentially 4-connected planar graph $G$ on $n$ vertices contains a…