Exponentially many Z5-colorings in simple planar graphs

Fractional coloring of triangle-free planar graphs

We prove that every planar triangle-free graph on $n$ vertices has fractional chromatic number at most $3-\frac{1}{n+1/3}$.

On edge-colorings of bicubic planar graphs

In the first part, we introduce a notion a degree of edge-colorings of bicubic plane graphs and proves some local formula of the graded number of colorings. In the second part, we give a new proof of a result of Fisk saying that any two edge-3-colorings of a planar bicubic graph are Kempe equivalent. Additionaly we show that the degree of colorings behaves well with respect to Kempe equivalence.

平面图的2距离着色

2-Distance Coloring of Planar Graphs with Specific Maximum Degree

研究最大度数至少为6的平面图的2距离着色问题，探讨其着色方法及结果。

3-coloring triangle-free planar graphs with a precolored 9-cycle

Given a triangle-free planar graph G and a 9-cycle C in G, we characterize situations where a 3-coloring of C does not extend to a proper 3-coloring of G. This extends previous results when C is a cycle of length at most 8.

Multiple DP-coloring of planar graphs without 3-cycles and normally adjacent 4-cycles

The concept of DP-coloring of a graph is a generalization of list coloring introduced by Dvořák and Postle in 2015. Multiple DP-coloring of graphs, as a generalization of multiple list coloring, was first studied by Bernshteyn, Kostochka and Zhu in 2019. This paper proves that planar graphs without 3-cycles and normally adjacent 4-cycles are $(7m, 2m)$-DP-colorable for every integer $m$. As a consequence, the strong fractional choice number of any planar graph without 3-cycles and normally adjacent 4-cycles is at most $7/2$.

Triangle-free planar graphs with at most $64^{n^{0.731}}$ 3-colorings

Thomassen conjectured that triangle-free planar graphs have exponentially many 3-colorings. Recently, he disproved his conjecture by providing examples of such graphs with $n$ vertices and at most $2^{15n/\log_2 n}$ 3-colorings. We improve his construction, giving examples of such graphs with at most $64^{n^{log_{9/2} 3}}<64^{n^{0.731}}$ 3-colorings. We conjecture this exponent is optimal.

A Unified Spiral Chain Coloring Algorithm for Planar Graphs

In this paper we have given a unified graph coloring algorithm for planar graphs. The problems that have been considered in this context respectively, are vertex, edge, total and entire colorings of the planar graphs. The main tool in the coloring algorithm is the use of spiral chain which has been used in the non-computer proof of the four color theorem in 2004. A more precies explanation of the proof of the four color theorem by spiral chain coloring is also given in this paper. Then we continue to spiral-chain coloring solutions by giving the proof of other famous conjectures of Vizing's to…

Strong edge coloring of planar graphs

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.

Improved square coloring of planar graphs

Square coloring is a variant of graph coloring where vertices within distance two must receive different colors. When considering planar graphs, the most famous conjecture (Wegner, 1977) states that $\frac32Δ+1$ colors are sufficient to square color every planar graph of maximum degree $Δ$. This conjecture has been proven asymptotically for graphs with large maximum degree. We consider here planar graphs with small maximum degree and show that $2Δ+7$ colors are sufficient, which improves the best known bounds when $6\leqslant Δ\leqslant 31$.

1-平面图中的公平着色

Equitable Coloring in 1-Planar Graphs

本文证明最大度不超过$r$的1-平面图在$r\ge13$时均可进行公平$r$-着色，推广了公平着色相关结论。

DP-3-coloring of some planar graphs

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.

An update on reconfiguring $10$-colorings of planar graphs

The reconfiguration graph $R_k(G)$ for the $k$-colorings of a graph $G$ has as vertex set the set of all possible proper $k$-colorings of $G$ and two colorings are adjacent if they differ in the color of exactly one vertex. A result of Bousquet and Perarnau (2016) regarding graphs of bounded degeneracy implies that if $G$ is a planar graph with $n$ vertices, then $R_{12}(G)$ has diameter at most $6n$. We improve on the number of colors, showing that $R_{10}(G)$ has diameter at most $8n$ for every planar graph $G$ with $n$ vertices.

不含某些环的平面图的松弛DP-3着色

Relaxed DP-3-coloring of planar graphs without some cycles

该研究属于图论着色领域，在已有松弛DP着色的平面图相关结论基础上，证明了不含4环与6环、或不含4环与8环的平面图满足DP-(0, 2, 2)-可着色性，拓展了该方向的研究结果。

Colorings of oriented planar graphs avoiding a monochromatic subgraph

For a fixed simple digraph $F$ and a given simple digraph $D$, an $F$-free $k$-coloring of $D$ is a vertex-coloring in which no induced copy of $F$ in $D$ is monochromatic. We study the complexity of deciding for fixed $F$ and $k$ whether a given simple digraph admits an $F$-free $k$-coloring. Our main focus is on the restriction of the problem to planar input digraphs, where it is only interesting to study the cases $k \in \{2,3\}$. From known results it follows that for every fixed digraph $F$ whose underlying graph is not a forest, every planar digraph $D$ admits an $F$-free $2$-coloring, a…

Planar graphs have two-coloring number at most 8

We prove that the two-colouring number of any planar graph is at most 8. This resolves a question of Kierstead et al. [SIAM J. Discrete Math.~23 (2009), 1548--1560]. The result is optimal.

最大度为4和5的平面图的节俭着色

Frugal coloring in planar graphs with maximum degree 4 and 5

该研究针对最大度为4和5的平面图，证明其2-节俭着色分别可使用7种和10种颜色，是Wegner猜想的部分变体结果。

超越平面性的图的均匀着色

Equitable coloring of graphs beyond planarity

针对IC-平面图和NIC-平面图的均匀Δ-着色问题，本文将Zhang证明的IC-平面图阈值从Δ≥12降至Δ≥10、NIC-平面图阈值从Δ≥13降至Δ≥11，优化了两类图均匀着色的最大度条件。

Three-coloring triangle-free graphs on surfaces V. Coloring planar graphs with distant anomalies

We settle a problem of Havel by showing that there exists an absolute constant d such that if G is a planar graph in which every two distinct triangles are at distance at least d, then G is 3-colorable. In fact, we prove a more general theorem. Let G be a planar graph, and let H be a set of connected subgraphs of G, each of bounded size, such that every two distinct members of H are at least a specified distance apart and all triangles of G are contained in \bigcup{H}. We give a sufficient condition for the existence of a 3-coloring phi of G such that for every B\in H, the restriction of phi t…

关于1-平面图奇数着色的一个注记

A Note on Odd Colorings of 1-Planar Graphs

本文研究1-平面图的奇数着色问题，证明每个1-平面图都可进行奇数23-着色。

Additive colorings of planar graphs

An \emph{additive coloring} of a graph $G$ is an assignment of positive integers $\{1,2,...,k\}$ to the vertices of $G$ such that for every two adjacent vertices the sums of numbers assigned to their neighbors are different. The minimum number $k$ for which there exists an additive coloring of $G$ is denoted by $η(G)$. We prove that $η(G)\leqslant 468$ for every planar graph $G$. This improves a previous bound $η(G)\leqslant 5544$ due to Norin. The proof uses Combinatorial Nullstellensatz and coloring number of planar hypergrahs. We also demonstrate that $η(G)\leqslant 36$ for 3-colorable plan…

Complete colorings of planar graphs

In this paper, we study the achromatic and the pseudoachromatic numbers of planar and outerplanar graphs as well as planar graphs of girth 4 and graphs embedded on a surface. We give asymptotically tight results and lower bounds for maximal embedded graphs.

Equitable Colorings of Planar Graphs without Short Cycles

An \emph{equitable coloring} of a graph is a proper vertex coloring such that the sizes of every two color classes differ by at most 1. Chen, Lih, and Wu conjectured that every connected graph $G$ with maximum degree $Δ\geq 2$ has an equitable coloring with $Δ$ colors, except when $G$ is a complete graph or an odd cycle or $Δ$ is odd and $G=K_{Δ,Δ}.$ Nakprasit proved the conjecture holds for planar graphs with maximum degree at least 9. Zhu and Bu proved that the conjecture holds for every $C_3$-free planar graph with maximum degree at least 8 and for every planar graph without $C_4$ and $C_5$…

平面图的缺陷对应染色

Defective correspondence coloring of planar graphs

本文研究平面图的缺陷对应染色，证明存在非3-缺陷3-可对应的平面图，构造了1-缺陷3-可对应但非4-可对应的平面图，并证明所有外平面图均为3-缺陷2-对应可染且缺陷数最优。

Sufficient conditions on planar graphs to have a relaxed DP-$3$-colorability

It is known that DP-coloring is a generalization of a list coloring in simple graphs and many results in list coloring can be generalized in those of DP-coloring. In this work, we introduce a relaxed DP-coloring which is a generalization if a relaxed list coloring. We also shows that every planar graph $G$ without $4$-cycles or $6$-cycles is DP-$(k,d)^*$-colorable. It follows immediately that $G$ is $(k,d)^*$-choosable.

Coupon-Coloring and total domination in Hamiltonian planar triangulations

We consider the so-called coupon-coloring of the vertices of a graph where every color appears in every open neighborhood, and our aim is to determine the maximal number of colors in such colorings. In other words, every color class must be a total dominating set in the graph and we study the total domatic number of the graph. We determine this parameter in every maximal outerplanar graph, and show that every Hamiltonian maximal planar graph has domatic number at least two, partially answering a conjecture of Goddard and Henning.

Odd coloring of sparse graphs and planar graphs

An {\it odd $c$-coloring} of a graph is a proper $c$-coloring such that each non-isolated vertex has a color appearing an odd number of times on its neighborhood. This concept was introduced very recently by Petru\v sevski and \v Skrekovski and has attracted considerable attention. Cranston investigated odd colorings of graphs with bounded maximum average degree, and conjectured that every graph $G$ with $mad(G)\leq \frac{4c-4}{c+1}$ has an odd $c$-coloring for $c\geq 4$, and proved the conjecture for $c\in\{5, 6\}$. In particular, planar graphs with girth at least $7$ and $6$ have an odd $5$-…

围长至少为5的平面图的2-距离（$Δ+4$）着色

On 2-Distance ($Δ+4$)-coloring of planar graphs with girth at least five

该研究针对围长不小于5的平面图，证明了当最大度$Δ\geq 22$时，这类图可使用$Δ+4$种颜色完成2-距离着色，推进了平面图2-距离着色的相关理论研究。

关于最大度为三的平面图的Wegner 8-着色定理

On Wegner's 8-Coloring Theorem for Planar Graphs of Maximum Degree Three

该研究简化了Wegner猜想特殊情况（最大度≤3的平面图可8色距离2着色）的证明，重点简化了原证明中移除5-环这一技术难点部分。

Weak Dynamic Coloring of Planar Graphs

The \textit{$k$-weak-dynamic number} of a graph $G$ is the smallest number of colors we need to color the vertices of $G$ in such a way that each vertex $v$ of degree $d(v)$ sees at least $\rm{min}\{k,d(v)\}$ colors on its neighborhood. We use reducible configurations and list coloring of graphs to prove that all planar graphs have 3-weak-dynamic number at most 6.

最大度不超过5的平面图的2-距离16染色

On 2-distance 16-coloring of planar graphs with maximum degree at most five

针对最大度不超过5的平面图的2-距离染色问题，证明其可进行2-距离16染色，改进了Hou等人2023年的相关研究结果。

List-coloring the Squares of Planar Graphs without 4-Cycles and 5-Cycles

Let $G$ be a planar graph without 4-cycles and 5-cycles and with maximum degree $Δ\ge 32$. We prove that $χ_{\ell}(G^2)\le Δ+3$. For arbitrarily large maximum degree $Δ$, there exist planar graphs $G_Δ$ of girth 6 with $χ(G_Δ^2)=Δ+2$. Thus, our bound is within 1 of being optimal. Further, our bound comes from coloring greedily in a good order, so the bound immediately extends to online list-coloring. In addition, we prove bounds for $L(p,q)$-labeling. Specifically, $λ_{2,1}(G)\le Δ+8$ and, more generally, $λ_{p,q}(G)\le (2q-1)Δ+6p-2q-2$, for positive integers $p$ and $q$ with $p\ge q$. Again, …

平面图的2距离(2Δ+7)着色

A 2-distance $(2Δ+7)$-coloring of planar graphs

本文改进了平面图2距离着色的色数结果，证明所有平面图均可使用2Δ+7种颜色进行2距离着色。

平面图中的全染色扩展

Extending total colorings in planar graphs

本文开创平面图全染色扩展研究，证明匹配子图在Δ≥28时可用Δ+3色扩展，距离3团在Δ≥27时可用Δ+1色扩展，且相应条件均不可再改进。

Acyclic edge-coloring of planar graphs: $Δ$ colors suffice when $Δ$ is large

An \emph{acyclic edge-coloring} of a graph $G$ is a proper edge-coloring of $G$ such that the subgraph induced by any two color classes is acyclic. The \emph{acyclic chromatic index}, $χ'_a(G)$, is the smallest number of colors allowing an acyclic edge-coloring of $G$. Clearly $χ'_a(G)\ge Δ(G)$ for every graph $G$. Cohen, Havet, and Müller conjectured that there exists a constant $M$ such that every planar graph with $Δ(G)\ge M$ has $χ'_a(G)=Δ(G)$. We prove this conjecture.

用无彼得森子图的4-和5-正则投影平面图进行边着色

Edge-coloring 4- and 5-regular projective planar graphs with no Petersen-minor

本文证明了无彼得森子图的4-和5-正则投影平面图可以被r-边着色。

Do triangle-free planar graphs have exponentially many 3-colorings?

Thomassen conjectured that triangle-free planar graphs have an exponential number of $3$-colorings. We show this conjecture to be equivalent to the following statement: there exists a positive real $α$ such that whenever $G$ is a planar graph and $A$ is a subset of its edges whose deletion makes $G$ triangle-free, there exists a subset $A'$ of $A$ of size at least $α|A|$ such that $G-(A\setminus A')$ is $3$-colorable. This equivalence allows us to study restricted situations, where we can prove the statement to be true.

Borowiecki-Broere 广义全染色猜想在平面图上的证明

The Borowiecki-Broere Generalized Total Coloring Conjecture for Planar Graphs

本文证明了每个有限简单平面图均存在四色广义全染色，即正常四顶点染色可扩展为四色森林边染色，且边色异于端点色，证明结合三角剖分计数不等式与拟阵划分定理，并给出 Lean 4 形式化。

Circular Coloring and Fractional Coloring in Planar Graphs

We study the following Steinberg-type problem on circular coloring: for an odd integer $k\ge 3$, what is the smallest number $f(k)$ such that every planar graph of girth $k$ without cycles of length from $k+1$ to $f(k)$ admits a homomorphism to the odd cycle $C_k$ (or equivalently, is circular $(k,\frac{k-1}{2})$-colorable). Known results and counterexamples on Steinberg's Conjecture indicate that $f(3)\in\{6,7\}$. In this paper, we show that $f(k)$ exists if and only if $k$ is an odd prime. Moreover, we prove that for any prime $p\ge 5$, $$p^2-\frac{5}{2}p+\frac{3}{2}\le f(p)\le 2p^2+2p-5.$$ …

Defective acyclic colorings of planar graphs

This paper studies two variants of defective acyclic coloring of planar graphs. For a graph $G$ and a coloring $φ$ of $G$, a 2CC transversal is a subset $E'$ of $E(G)$ that intersects every 2-colored cycle. Let $k$ be a positive integer. We denote by $m_k(G)$ the minimum integer $m$ such that $G$ has a proper $k$-coloring which has a 2CC transerval of size $m$, and by $m'_k(G)$ the minimum size of a subset $E'$ of $E(G)$ such that $G-E'$ is acyclic $k$-colorable. We prove that for any $n$-vertex $3$-colorable planar graph $G$, $m_3(G) \le n - 3$ and for any planar graph $G$, $m_4(G) \le n - 5$…

平面图6-围长2-距离着色的改进上界

An improved bound for 2-distance coloring of planar graphs with girth six

针对围长至少为6且最大度$Δ\neq6$的平面图，证明其2-距离色数$χ_2(G)\neqΔ+4$，改进了该类图2-距离着色的已知最优上界。

最大度为6的平面图的2-距离20染色

2-distance 20-coloring of planar graphs with maximum degree 6

本文证明最大度至多6的平面图具有2-距离20染色，将已知上界从21改进至20。

最大度为5的平面图的改进2-距离染色

Improved 2-Distance Coloring of Planar Graphs with Maximum Degree 5

本文研究了平面图的2-距离染色问题，证明了最大度不超过5的平面图的2-距离色数上界可由18改进至17。

Tree-colorable maximal planar graphs

A tree-coloring of a maximal planar graph is a proper vertex $4$-coloring such that every bichromatic subgraph, induced by this coloring, is a tree. A maximal planar graph $G$ is tree-colorable if $G$ has a tree-coloring. In this article, we prove that a tree-colorable maximal planar graph $G$ with $δ(G)\geq 4$ contains at least four odd-vertices. Moreover, for a tree-colorable maximal planar graph of minimum degree 4 that contains exactly four odd-vertices, we show that the subgraph induced by its four odd-vertices is not a claw and contains no triangles.

Defective 2-colorings of planar graphs without 4-cycles and 5-cycles

Let $G$ be a graph without 4-cycles and 5-cycles. We show that the problem to determine whether $G$ is $(0,k)$-colorable is NP-complete for each positive integer $k.$ Moreover, we construct non-$(1,k)$-colorable planar graphs without 4-cycles and 5-cycles for each positive integer $k.$ Finally, we prove that $G$ is $(d_1,d_2)$-colorable where $(d_1,d_2)=(4,4), (3,5),$ and $(2,9).$

4-非正则平面图的2-距离着色

2-Distance Coloring of 4-Irregular Planar Graphs

本文研究4-非正则平面图的2-距离着色问题，将其2-距离色数上界从Zhu证明的13改进为10，推进了平面图着色相关理论的研究。

Coloring count cones of planar graphs

For a plane near-triangulation $G$ with the outer face bounded by a cycle $C$, let $n^\star_G$ denote the function that to each $4$-coloring $ψ$ of $C$ assigns the number of ways $ψ$ extends to a $4$-coloring of $G$. The block-count reducibility argument (which has been developed in connection with attempted proofs of the Four Color Theorem) is equivalent to the statement that the function $n^\star_G$ belongs to a certain cone in the space of all functions from $4$-colorings of $C$ to real numbers. We investigate the properties of this cone for $|C|=5$, formulate a conjecture strengthening the…

平面图的B-着色研究

On B-Colorings in Planar Graphs

本文针对平面图的B-着色问题，改进了Kong等人的已有界，证明任意图G为平面图时，其B-着色所需最少颜色数不超过Δ+max{Δ₂,38}。

全染色猜想对不含三种特殊子图的平面图成立

The Total Coloring Conjecture holds for planar graphs without three special subgraphs

本文证明全染色猜想对最大度为六且不含蘑菇、帐篷和锥体三种子结构的平面图成立，从而推广了此前关于禁止4-扇的结果。

$k$-intersection edge-coloring subcubic planar multigraphs

Given an edge-coloring of a simple graph, assign to every vertex $v$ a set $S_v$ comprised of the colors used on the edges incident to $v$. The $k$-intersection chromatic index of a graph is the minimum $t$ such that the edge set can be properly $t$-colored, additionally requiring that for every two adjacent vertices $u$ and $v$, $|S_u \cap S_v| \le k$. For all $k \neq 2$, this value is known for subcubic planar graphs, and furthermore, these values are best possible. We naturally extend this definition to multigraphs with bounded edge multiplicity, and we show that every subcubic planar multi…

DP-4-colorability of two classes of planar graphs

DP-coloring (also known as correspondence coloring) is a generalization of list coloring introduced recently by Dvořák and Postle (2017). In this paper, we prove that every planar graph $G$ without $4$-cycles adjacent to $k$-cycles is DP-$4$-colorable for $k=5$ and $6$. As a consequence, we obtain two new classes of $4$-choosable planar graphs. We use identification of verticec in the proof, and actually prove stronger statements that every pre-coloring of some short cycles can be extended to the whole graph.

关于围长为5的平面图2-距离着色的若干结果

Some results on 2-distance coloring of planar graphs with girth five

针对围长为5的平面图的2-距离着色问题，本文证明其存在2-距离$Δ+7$着色（改进已有结果），且当最大度$Δ\geq 10$时存在2-距离$Δ+6$着色，为该类图的着色研究提供了更优结论。

A Thomassen-type method for planar graph recoloring

The reconfiguration graph $R_k(G)$ for the $k$-colorings of a graph $G$ has as vertices all possible $k$-colorings of $G$ and two colorings are adjacent if they differ in the color of exactly one vertex. We use a list coloring technique inspired by results of Thomassen to prove that for a planar graph $G$ with $n$ vertices, $R_{10}(G)$ has diameter at most $8n$, and if $G$ is triangle-free, then $R_7(G)$ has diameter at most $7n$.

Bounds for the game coloring number of planar graphs with a specific girth

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…

关于平面图稳健染色的注记

Note on robust coloring of planar graphs

本文研究平面图的稳健色数，通过构造一个满足 $χ_1(G) = 3$ 的无限平面图族，解决了 Bacsó 等人近期提出的关于稳健染色的一个问题。