Related papers: Local girth choosability of planar graphs

Grotzsch proved that every triangle-free planar graph is 3-colorable. Thomassen proved that every planar graph of girth at least five is 3-choosable. As for other surfaces, Thomassen proved that there are only finitely many 4-critical…

Thomassen showed that planar graphs are 5-list-colourable, and that planar graphs of girth at least five are 3-list-colourable. An easy degeneracy argument shows that planar graphs of girth at least four are 4-list-colourable. In 2022,…

Thomassen proved that any plane graph of girth 5 is list-colorable from any list assignment such that all vertices have lists of size two or three and the vertices with list of size two are all incident with the outer face and form an…

One of Thomassen's classical results is that every planar graph of girth at least $5$ is 3-choosable. One can wonder if for a planar graph $G$ of girth sufficiently large and a $3$-list-assignment $L$, one can do even better. Can one find…

Thomassen proved that every planar graph $G$ on $n$ vertices has at least $2^{n/9}$ distinct $L$-colorings if $L$ is a 5-list-assignment for $G$ and at least $2^{n/10000}$ distinct $L$-colorings if $L$ is a 3-list-assignment for $G$ and $G$…

In 1994, Thomassen proved that every planar graph is 5-list-colorable. In 1995, Thomassen proved that every planar graph of girth at least five is 3-list-colorable. His proofs naturally lead to quadratic-time algorithms to find such…

Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G has girth at least six and all lists have size at least three, then there exists an L-coloring respecting at…

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…

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 \emph{signed graph} is a pair $\Gs$ in which $G$ is a finite simple graph and $\sigma:\E(G)\to\{+1,-1\}$ is a \emph{signature}. Following M\'a\v{c}ajov\'a--Raspaud- \v{S}koviera and Jin--Kang--Steffen, a \emph{proper coloring} of $\Gs$ is…

A graph G is k-choosable if G can be properly colored whenever every vertex has a list of at least k available colors. Thomassen's theorem states that every planar graph is 5-choosable. We extend the result by showing that every graph with…

Proper graph coloring assigns different colors to adjacent vertices of the graph. Usually, the number of colors is fixed or as small as possible. Consider applications (e.g. variants of scheduling) where colors represent limited resources…

Thomassen (1994) proved that every planar graph is 5-choosable. This result was generalised by {\v{S}}krekovski (1998) and He et al. (2008), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the…

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…

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…

We show that there exists a constant $c > 0$ such that if $G$ is a planar graph with 5-correspondence assignment $(L,M)$, then $G$ has at least $2^{c\cdot v(G)}$ distinct $(L,M)$-colourings. This confirms a conjecture of Langhede and…

Let $G$ be a $\{C_4, C_5\}$-free planar graph with a list assignment $L$. Suppose a preferred color is given for some of the vertices. We prove that if all lists have size at least four, then there exists an $L$-coloring respecting at least…

Thomassen proved that all planar graphs are $5$-choosable. \v{S}krekovski strengthened the result by showing that all $K_{5}$-minor-free graphs are $5$-choosable. Dvo\v{r}\'{a}k and Postle pointed out that all planar graphs are…

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…

Let $G$ be a planar embedding with list-assignment $L$ and outer cycle $C$, and let $P$ be a path of length at most four on $C$, where each vertex of $G\setminus C$ has a list of size at least five and each vertex of $C\setminus P$ has a…