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 girth at least five is 3-choosable. In this paper, we introduce the concept of a \emph{local girth list assignment}: a list assignment wherein the list size of a vertex depends not on the girth of the graph, but rather on the length of the shortest cycle in which the vertex is contained. We give a local list colouring theorem unifying the two theorems of Thomassen mentioned above. In particular, we show that if $G$ is a planar graph and $L$ is a list assignment for $G$ such that $|L(v)| \geq 3$ for all $v \in V(G)$; $|L(v)| \geq 4$ for every vertex $v$ contained in a 4-cycle; and $|L(v)| \geq 5$ for every $v$ contained in a triangle, then $G$ admits an $L$-colouring.
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 independent set. We present a strengthening of this result, relaxing the constraint on the vertices with list of size two. This result is used to bound the size of the 3-list-coloring critical plane graphs with one precolored face.
An injective coloring of a graph $G$ is an assignment of colors to the vertices of $G$ so that any two vertices with a common neighbor have distinct colors. A graph $G$ is injectively $k$-choosable if for any list assignment $L$, where $|L(v)| \geq k$ for all $v \in V(G)$, $G$ has an injective $L$-coloring. Injective colorings have applications in the theory of error-correcting codes and are closely related to other notions of colorability. In this paper, we show that subcubic planar graphs with girth at least 6 are injectively 5-choosable. This strengthens a result of Lužar, Škrekovski, and T…
Assume $ k $ is a positive integer, $ λ=\{k_1,k_2,...,k_q\} $ is a partition of $ k $ and $ G $ is a graph. A $λ$-assignment of $ G $ is a $ k $-assignment $ L $ of $ G $ such that the colour set $ \bigcup_{v\in V(G)} L(v) $ can be partitioned into $ q $ subsets $ C_1\cup C_2\cup\cdots\cup C_q $ and for each vertex $ v $ of $ G $, $ |L(v)\cap C_i|=k_i $. We say $ G $ is $λ$-choosable if for each $λ$-assignment $ L $ of $ G $, $ G $ is $ L $-colourable. In particular, if $ λ=\{k\} $, then $λ$-choosable is the same as $ k $-choosable, if $ λ=\{1, 1,...,1\} $, then $λ$-choosable is equivalent to …
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 $k \in \{3,4,5,6\}$, every signed planar graph without circuits of length $k$ is 4-choosable. Furthermore, every signed planar graph without circuits of length 3 and of length 4 is 3-choosable. We construct a signed planar graph with girth 4 which is not 3-choosable but the unsigned graph is 3-choosable.
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 xy, the lists L(x) and L(y) share at most d colors. A graph G is (k,d)-choosable if there exists an L-coloring of G for every (k,d)-list assignment L. This concept is also known as choosability with separation. We prove that planar graphs without 4-cycles are (3,1)-choosable and that planar graphs without 5-cycles and 6-cycles are (3,1)-choosable. In addition…
We study squares of planar graphs with the aim to determine their list chromatic number. We present new upper bounds for the square of a planar graph with maximum degree $Δ\leq 4$. In particular $G^2$ is 5-, 6-, 7-, 8-, 12-, 14-choosable if the girth of $G$ is at least 16, 11, 9, 7, 5, 3 respectively. In fact we prove more general results, in terms of maximum average degree, that imply the results above.
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 given any list assignment $L$ to the vertices of $G$, with $|L(v)| \ge k$ for all $v$, and any edge-coloring $F$ of $G$, $G$ admits a coloring $c$ adapted to $F$ where $c(v) \in L(v)$ for all $v$, then $G$ is said to be \emph{adaptably $k$-choosable}. A {\em $(k,d)$-list assignment} for a graph $G$ is a map that assigns to each vertex $v$ a list $L(v)$ of at le…
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 applying the famous Combinatorial Nullstellensatz, we design an effective algorithm to deal with list coloring problems. At the same time, we prove that a planar graph $G$ is $4$-choosable if any two $4$-cycles having distance at least $5$ in $G$, which extends the result of Lam et al.
In this paper, we study the concept of edge-group choosability of graphs. We say that G is edge k-group choosable if its line graph is k-group choosable. An edge-group choosability version of Vizing conjecture is given. The evidence of our claim are graphs with maximum degree less than 4, planar graphs with maximum degree at least 11, planar graphs without small cycles, outerplanar graphs and near-outerplanar graphs.
Let $G$ be a planar graph with no two 3-cycles sharing an edge. We show that if $Δ(G)\geq 9$, then $χ'_l(G) = Δ(G)$ and $χ''_l(G)=Δ(G)+1.$ We also show that if $Δ(G)\geq 6$, then $χ'_l(G)\leqΔ(G)+1$ and if $Δ(G)\geq 7$, then $χ''_l(G)\leqΔ(G)+2$. All of these results extend to graphs in the projective plane and when $Δ(G)\geq 7$ the results also extend to graphs in the torus and Klein bottle. This second edge-choosability result improves on work of Wang and Lih and of Zhang and Wu. All of our results use the discharging method to prove structural lemmas about the existence of subgraphs with sm…
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 Grötzsch's result that triangle-free planar graphs are such, most of the effort was focused to solving Havel's and Steinberg's conjectures. In this paper, we prove that every planar graph of maximum degree $4$ obtained as a subgraph of the medial graph of any bipartite plane graph is $3$-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence represen…
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 of constraining the structure of the graph, for any $\ell \in \{3,4,5,6,7\}$, a planar graph is 4-choosable if it is $\ell$-cycle-free. In terms of constraining the list assignment, one refinement of $k$-choosability is choosability with separation. A graph is $(k,s)$-choosable if the graph is colorable from lists of size $k$ where adjacent vertices have at…
It is known that every loopless cubic graph is 4-edge choosable. We prove the following strengthened result. Let G be a planar cubic graph having b cut-edges. There exists a set F of at most 5b/2 edges of G with the following property. For any function L which assigns to each edge of F a set of 4 colours and which assigns to each edge in E(G)-F a set of 3 colours, the graph G has a proper edge colouring where the colour of each edge e belongs to L(e).
We show that the choosability of the square of planar graphs of max degree 4 without five cycles is at most 12. Keywords: planar graph, choosability AMS Mathematics Subject Classification: 05C15
In this paper, we aim to introduce the group version of edge coloring and list edge coloring, and prove that all 2-degenerate graphs along with some planar graphs without adjacent short cycles is group $(Δ(G)+1)$-edge-choosable while some planar graphs with large girth and maximum degree is group $Δ(G)$-edge-choosable.
We show that every graph with two crossings is 5-choosable. We also prove that every graph which can be made planar by removing one edge is 5-choosable.
An $r$-augmented tree is a rooted tree plus $r$ edges added from each leaf to ancestors. For $d,g,r\in\mathbb{N}$, we construct a bipartite $r$-augmented complete $d$-ary tree having girth at least $g$. The height of such trees must grow extremely rapidly in terms of the girth. Using the resulting graphs, we construct sparse non-$k$-choosable bipartite graphs, showing that maximum average degree at most $2(k-1)$ is a sharp sufficient condition for $k$-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-$k$-colorable graphs and hyperg…
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 every $(\ell, k)$-list assignment $L$. We prove that if $G$ is a planar graph with $(4,2)$-list assignment $L$ and for every triangle $T \subseteq G$ we have that $|\bigcap_{v \in V(T)} L(v)| \neq 2$, then $G$ is $L$-colourable. In fact, we prove a slightly stronger result: if $G$ contains a clique $H$ such that $V(H) \cap V(T) \neq \emptyset$ for every triangl…
We generalize a framework of list colouring results to correspondence colouring. Correspondence colouring is a generalization of list colouring wherein we localize the meaning of the colours available to each vertex. As pointed out by Dvořák and Postle, both of Thomassen's theorems on the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five carry over to the correspondence colouring setting. In this paper, we show that the family of graphs that are critical for 5-correspondence colouring as well as the family of graphs of girth at least five that are criti…
Thomassen proved that all planar graphs are $5$-choosable. Škrekovski strengthened the result by showing that all $K_{5}$-minor-free graphs are $5$-choosable. Dvořák and Postle pointed out that all planar graphs are DP-$5$-colorable. In this note, we first improve these results by showing that every $K_{5}$-minor-free or $K_{3, 3}$-minor-free graph is DP-$5$-colorable. In the final section, we further improve these results under the term strictly $f$-degenerate transversal.
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 $π$ that assigns a color $π(v)\in L(v)$ to each vertex $v\in V(G)$ so that at most $d$ neighbors of $v$ receive color $π(v)$. A graph $G$ is said to be $(k,d)^*$-choosable if it admits an $(L,d)^*$-coloring for every list assignment $L$ with $|L(v)|\ge k$ for all $v\in V(G)$. In 2001, Lih et al. \cite{LSWZ-01} proved that planar graphs without 4- and $l$-cycles are $(3,1)^*$-choosable, where $l\in \{5,6,7\}$. Later, Dong and Xu \cite{DX-09} prov…
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 to the problem of Havel regarding 3-colorability of planar graphs with triangles far apart.
Assume $k$ is a positive integer, $λ=\{k_1, k_2, \ldots, k_q\}$ is a partition of $k$ and $G$ is a graph. A $λ$-list assignment of $G$ is a $k$-list assignment $L$ of $G$ such that the colour set $\cup_{v\in V(G)}L(v)$ can be partitioned into $q$ subsets $C_1 \cup C_2 \ldots \cup C_q$ and for each vertex $v$ of $G$, $|L(v) \cap C_i| \ge k_i$. We say $G$ is $λ$-choosable if for each $λ$-list assignment $L$ of $G$, $G$ is $L$-colourable. It follows from the definition that if $λ=\{k\}$, then $λ$-choosable is the same as $k$-choosable, if $λ=\{1,1,\ldots, 1\}$, then $λ$-choosable is equivalent to…
We study the linear list chromatic number, denoted $\lcl(G)$, of sparse graphs. The maximum average degree of a graph $G$, denoted $\mad(G)$, is the maximum of the average degrees of all subgraphs of $G$. It is clear that any graph $G$ with maximum degree $Δ(G)$ satisfies $\lcl(G)\ge \ceil{Δ(G)/2}+1$. In this paper, we prove the following results: (1) if $\mad(G)<12/5$ and $Δ(G)\ge 3$, then $\lcl(G)=\ceil{Δ(G)/2}+1$, and we give an infinite family of examples to show that this result is best possible; (2) if $\mad(G)<3$ and $Δ(G)\ge 9$, then $\lcl(G)\le\ceil{Δ(G)/2}+2$, and we give an infinite…
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 L-coloring of G for every (k,d)-list assignment L. This concept is also known as choosability with separation. It is known that planar graphs are (4,1)-choosable but it is not known if planar graphs are (3,1)-choosable. We strengthen the result that planar graphs are (4,1)-choosable by allowing an independent set of vertices to have lists of size 3 instead …
For a planar graph $G$, Borodin stated that $G$ is $(Δ+1)$-edge-choosable if $Δ\geq9$ and later Bonamy showed that $G$ is $9$-edge-choosable if $Δ=8$. At the same time, Borodin et al. proved that $G$ is $Δ$-edge-choosable if $Δ\geq12$. In the paper, we extend these results to $K_5$-minor free graphs.
In 1991, Bollobás and Frieze showed that the threshold for $G_{n,p}$ to contain a spanning maximal planar subgraph is very close to $p = n^{-1/3}$. In this paper, we compute similar threshold ranges for $G_{n,p}$ to contain a maximal bipartite planar subgraph and for $G_{n,p}$ to contain a maximal planar subgraph of fixed girth $g$.
We give a new proof of the fact that every planar graph is 5-choosable, and use it to show that every graph drawn in the plane so that the distance between every pair of crossings is at least 15 is 5-choosable. At the same time we may allow some vertices to have lists of size four only, as long as they are far apart and far from the crossings.
For a positive integer $k$, the $k$-recolouring graph of a graph $G$ has as vertex set all proper $k$-colourings of $G$ with two $k$-colourings being adjacent if they differ by the colour of exactly one vertex. A result of Dyer et al. regarding graphs of bounded degeneracy implies that the $7$-recolouring graphs of planar graphs, the $5$-recolouring graphs of triangle-free planar graphs and the $4$-recolouring graphs planar graphs of girth at least six are connected. On the other hand, there are planar graphs whose $6$-recolouring graph is disconnected, triangle-free planar graphs whose $4$-re…
A graph is called hypohamiltonian if it is not hamiltonian but becomes hamiltonian if any vertex is removed. Many hypohamiltonian planar cubic graphs have been found, starting with constructions of Thomassen in 1981. However, all the examples found until now had 4-cycles. In this note we present the first examples of hypohamiltonian planar cubic graphs with cyclic connectivity five, and thus girth five. We show by computer search that the smallest members of this class are three graphs with 76 vertices.
Degree-choosability of proper conflict-free list coloring of sparse graphs
Faculty of Science and Technology, Keio University（庆应义塾大学理工学部）；Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）；Faculty of Information Studies, Novo Mesto（新梅斯托信息学院）；Rudolfovo - Science and Technology Centre Novo Mesto（鲁多尔福沃-新梅斯托科学与技术中心）；School of Mathematical and Science, Zhejiang Normal University（浙江师范大学数理学院）
This paper proves that the Alon-Tarsi number of any planar graph is at most $5$, which gives an alternate proof of the $5$-choosability as well as the $5$-paintability of planar graphs.
A graph is $(d_1, \ldots, d_k)$-colorable if its vertex set can be partitioned into $k$ nonempty subsets so that the subgraph induced by the $i$th part has maximum degree at most $d_i$ for each $i\in\{1, \ldots, k\}$. It is known that for each pair $(d_1, d_2)$, there exists a planar graph with girth $4$ that is not $(d_1, d_2)$-colorable. This sparked the interest in finding the pairs $(d_1, d_2)$ such that planar graphs with girth at least $5$ are $(d_1, d_2)$-colorable. Given $d_1\leq d_2$, it is known that planar graphs with girth at least $5$ are $(d_1, d_2)$-colorable if either $d_1\geq …
We prove a conjecture of Dvořák, Král, Nejedlý, and Škrekovski that planar graphs of girth at least five are square $(Δ+2)$-colorable for large enough $Δ$. In fact, we prove the stronger statement that such graphs are square $(Δ+2)$-choosable and even square $(Δ+2)$-paintable.
List coloring generalizes graph coloring by requiring the color of a vertex to be selected from a list of colors specific to that vertex. One refinement of list coloring, called choosability with separation, requires that the intersection of adjacent lists is sufficiently small. We introduce a new refinement, called choosability with union separation, where we require that the union of adjacent lists is sufficiently large. For $t \geq k$, a $(k,t)$-list assignment is a list assignment $L$ where $|L(v)| \geq k$ for all vertices $v$ and $|L(u)\cup L(v)| \geq t$ for all edges $uv$. A graph is $(k…
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 first show that for every triangle-free planar graph G and a vertex x of G, the graph G has a set coloring phi by subsets of {1,...,6} such that |phi(v)|>=2 for each vertex v of G and |phi(x)|=3. As a corollary, every triangle-free planar graph on n vertices is (6n:2n+1)-colorable. We further use this result to prove that for every Delta, there exists a constant M_Delta such that every planar graph G of girth at least five a…
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.
A feedback vertex set of a graph is a subset of vertices intersecting all cycles. We provide tight upper bounds on the size of a minimum feedback vertex set in planar graphs of girth at least five. We prove that if $G$ is a connected planar graph of girth at least five on $n$ vertices and $m$ edges, then $G$ has a feedback vertex set of size at most $\frac{2m-n+2}{7}$. By Euler's formula, this implies that $G$ has a feedback vertex set of size at most $\frac{m}{5}$ and $\frac{n-2}{3}$. These results not only improve a result of Dross, Montassier and Pinlou and confirm the girth-5 case of one o…
Thomassen (1994) proved that every planar graph is 5-choosable. This result was generalised by {Š}krekovski (1998) and He et al. (2008), who proved that every $K_5$-minor-free graph is 5-choosable. Both proofs rely on the characterisation of $K_5$-minor-free graphs due to Wagner (1937). This paper proves the same result without using Wagner's structure theorem or even planar embeddings. Given that there is no structure theorem for graphs with no $K_6$-minor, we argue that this proof suggests a possible approach for attacking the Hadwiger Conjecture.
A strong $k$-edge-coloring of a graph $G$ is a mapping from $E(G)$ to $\{1,2,\ldots,k\}$ such that every pair of distinct edges at distance at most two receive different colors. The strong chromatic index $χ'_s(G)$ of a graph $G$ is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. Denote $σ(G)=\max_{xy\in E(G)}\{\operatorname{deg}(x)+\operatorname{deg}(y)-1\}$. It is easy to see that $σ(G) \le χ'_s(G)$ for any graph $G$, and the equality holds when $G$ is a tree. For a planar graph $G$ of maximum degree $Δ$, it was proved that $χ'_s(G) \le 4 Δ+4$ by using the Four Color Theorem. T…
Let ${\rm col_g}(G)$ be the game coloring number of a given graph $G.$ Define the game coloring number of a family of graphs $\mathcal{H}$ as ${\rm col_g}(\mathcal{H}) := \max\{{\rm col_g}(G):G \in \mathcal{H}\}.$ Let $\mathcal{P}_k$ be the family of planar graphs of girth at least $k.$ We show that ${\rm col_g}(\mathcal{P}_7) \leq 5.$ This result extends a result about the coloring number by Wang and Zhang {WZ11} (${\rm col_g}(\mathcal{P}_8) \leq 5).$ We also show that these bounds are sharp by constructing a graph $G$ where $G \in {\rm col_g}(\mathcal{P}_k) \geq 5$ for each $k \leq 8$ such t…
A strong edge coloring of a graph is a proper edge coloring where the edges at distance at most two receive distinct colors. It is known that every planar graph with maximum degree D has a strong edge coloring with at most 4D + 4 colors. We show that 3D + 6 colors suffice if the graph has girth 6, and 3D colors suffice if the girth is at least 7. Moreover, we show that cubic planar graphs with girth at least 6 can be strongly edge colored with at most 9 colors.
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 $|L(u)\cap L(v)|\le c$ for any edge $uv$ of $G$, there exists an assignment $φ$ of sets of integers to the vertices of $G$ such that $φ(u)\subset L(u)$ and $|φ(v)|=b$ for any vertex $v$ and $φ(u)\cap φ(v)=\emptyset$ for any edge $uv$. Such a value of $c$ is called the separation number of $(G,a,b)$. We also study the variant called the free-separation number which is d…
We consider 3-regular planar matchstick graphs, i.e. those which have a planar embedding such that all edge lengths are equal, with given girth g. For girth 3 it is known that such graphs exist if and only if the number of vertices n is an even integer larger or equal to 8. Here we prove that such graphs exist for girth g=4 if and only if n is even and at least 20. We provide an example for girth g=5 consisting of 180 vertices.
We prove that planar graphs of maximum degree 3 and of girth at least 7 are 3-edge-colorable, extending the previous result for girth at least 8 by Kronk, Radlowski, and Franen from 1974.
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 graphs of girth at least five embeddable in any fixed surface. This implies a linear-time algorithm for deciding 3-colorablity for graphs of girth at least five on any fixed surface. Dvorak, Kral and Thomas strengthened Thomassen's result by proving that the number of vertices in a 4-critical graph of girth at least five is linear in its genus. They used this res…
We introduce a new variant of graph coloring called correspondence coloring which generalizes list coloring and allows for reductions previously only possible for ordinary coloring. Using this tool, we prove that excluding cycles of lengths 4 to 8 is sufficient to guarantee 3-choosability of a planar graph, thus answering a question of Borodin.
