On planar regular graphs degree three without Hamiltonian cycles
Emanuels Grinbergs
给出了可平面图存在哈密顿圈的必要条件。给出了不含哈密顿圈的三度正则可平面图的例子。
Necessary condition to have Hamiltonian cycle in planar graph is given. Examples of regular planar graphs degree three without Hamiltonian cycle are built.
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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 DP-$4$-colorable. Recently, Kim and Yu extended the result on $3$- and $4$-cycles by showing that planar graphs without triangles adjacent to $4$-cycles are DP-$4$-colorable. Xu and Wu showed that planar graphs without $5$-cycles adjacent simultaneously to $3$-cycles and $4$-cycles are $4$-choosable. In this paper, we extend the result on $5$-cycles and triangles adja…
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 any chosen $i,$ if every $i$-cycle of a planar graph $G$ is not simultaneously adjacent to $j$-cycles, $k$-cycles, and $l$-cycles, then $G$ is $4$-choosable.
Generating the cycle space of planar graphs
证明了平面可分3-连通图$G$的圈空间是由一个${\rm Aut}(G)$ -不变圈套集生成的。我们还讨论了在较小的连通性的情况下的情况。
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
This paper proves that if $G$ is a planar graph without 4-cycles and $l$-cycles for some $l\in\{5, 6, 7\}$, then there exists a matching $M$ such that $AT(G-M)\leq 3$. This implies that every planar graph without 4-cycles and $l$-cycles for some $l\in\{5, 6, 7\}$ is 1-defective 3-paintable.
This paper proves that every planar graph without cycles of length 4, 7, or 9 is DP-3-colorable.
Whitney proved in 1931 that every 4-connected planar triangulation is hamiltonian. Later in 1979, Hakimi, Schmeichel and Thomassen conjectured that every such triangulation on $n$ vertices has at least $2(n - 2)(n - 4)$ hamiltonian cycles. Along this direction, Brinkmann, Souffriau and Van Cleemput established a linear lower bound on the number of hamiltonian cycles in 4-connected planar triangulations. In stark contrast, Alahmadi, Aldred and Thomassen showed that every 5-connected triangulation of the plane or the projective plane has exponentially many hamiltonian cycles. This gives the moti…
DP-coloring (also known as correspondence coloring) of a simple graph is a generalization of list coloring. It is known that planar graphs without 4-cycles adjacent to triangles are 4-choosable, and planar graphs without 4-cycles are DP-4-colorable. In this paper, we show that planar graphs without 4-cycles adjacent to triangles are DP-4-colorable, which is an extension of the two results above.
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 $7/2$.
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 that if a planar graph $G$ satisfies one of the these conditions, then $G$ is 4-choosable. This yields that if each 5-cycle of a planar graph $G$ is not adjacent a 3-cycle, then $G$ is 4-choosable.
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 $(2,9).$
Local fractional metric dimension of rotationally symmetric planar graphs arisen from planar chorded cycles
本文通过平面弦环的边并合构造新的旋转对称平面图族，确定其局部分数度量维数，并利用渐近行为证明存在常数或有界局部分数维数的图族。
An algorithm for finding Hamiltonian Cycles in Cubic Planar Graphs
首先证明了在三次可平面图中寻找哈密顿圈与在对偶图中寻找具有特定性质的树之间的一一对应关系。利用这些信息，我们构造了一个精确的算法，在三次可平面图中寻找哈密顿圈。我们的算法的最坏情况下的时间复杂度是O $(2^n)$。
The 3-colorability of planar graphs without cycles of length 4, 6 and 9
本文证明了不含长为4，6，9的圈的可平面图是3-可着色的。
A comment to: On 3-colorable planar graphs without short cycles
设G是一个图。证明了：如果G是一个不含{4，6，7}-圈且两个5-圈不共用一条边的可平面图，则G是3-可着色的.我们注意到这个结果的证明是不正确的。
We prove that, for every natural number $k$, every sufficiently large 3-connected cubic planar graph has a cycle whose length is in $[k,2k+9]$. We also show that this bound is close to being optimal by constructing, for every even $k\geq 4$, an infinite family of 3-connected cubic planar graphs that contain no cycle whose length is in $[k,2k+1]$.
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.
Every planar graph without adjacent short cycles is 3-colorable
如果两个循环具有共同的边沿，则它们是{ \em相邻}的。设$G$是一个可平面图，对于任意两个相邻的圈$C_{1}$和$C_{2}$，我们有$|C_{1}| + |C_{2}| \geq 11$，特别地，当$|C_{1}| = 5$，$|C_{2}| \geq 7$ .证明了图$G$是3-可着色的.
A graph is near-bipartite if its vertex set can be partitioned into an independent set and a set which induces a forest. In this paper, planar graphs without cycles of length from 4 to 7 are shown to be near-bipartite.
Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we show that every $4$-connected planar triangulation on $n$ vertices has $Ω(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{Ω(n^{1/4})}$ Hamiltonian cycles.
In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the vertex set of a forest into two independent sets. However, a pair of independent sets combined may not induce a forest. Thus our result extends the result of Wang and Xu (2013) stating that the vertex set of every planar graph without $4$-cycles and $6$-cycles can be partitioned into three sets, wher…
A generalization of list-coloring, now known as DP-coloring, was recently introduced by Dvořák and Postle. Essentially, DP-coloring assigns an arbitrary matching between lists of colors at adjacent vertices, as opposed to only matching identical colors as is done for list-coloring. Several results on list-coloring of planar graphs have since been extended to the setting of DP-coloring. We note that list-coloring results do not always extend to DP-coloring results. Our main result in this paper is to prove that every planar graph without cycles of length $\{4, a, b, 9\}$ for $a, b \in \{6, 7, 8…
Hu and Li investigate the signed graph version of Erd$\ddot{\mathrm{o}}$s problem: Is there a constant $c$ such that every signed planar graph without $k$-cycles, where $4\leq k\leq c$, is $3$-colorable and prove that each signed planar graph without cycles of length from 4 to 8 is 3-colorable. We give a very short and simple proof of this result and improve it, based on a recent observation.
DP-coloring as a generalization of list coloring was introduced by Dvořák and Postle in 2017, who proved that every planar graph without cycles from 4 to 8 is 3-choosable, which was conjectured by Borodin {\it et al.} in 2007. In this paper, we prove that every planar graph without adjacent cycles of length at most $8$ is $3$-choosable, which extends this result of Dvořák and Postle.
We resolve a conjecture of Cox and Martin by determining asymptotically for every $k\ge 2$ the maximum number of copies of $C_{2k}$ in an $n$-vertex planar graph.
Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices then $G$ contains at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar $n$-vertex triangulations $G$ that do not have too many separating 4-cyc…
In the 1960s, Erdős and his cooperators initiated the research of the maximum numbers of edges in a graph or a planar graph on $n$ vertices without $k$ edge-disjoint cycles. This problem had been solved for $k\leq4$. As pointed out by Bollobás, it is very difficult for general $k$. Recently, Tait and Tobin [J. Combin. Theory Ser. B, 2017] confirmed a famous conjecture on maximum spectral radius of $n$-vertex planar graphs. Motivated by the above results, we consider two extremal problems on planar graphs without $k$ edge-disjoint cycles. We first determine the maximum number of edges in a plan…
DP-coloring (also known as correspondence coloring) is a generalization of list coloring, introduced by Dvořák and Postle in 2017. It is well-known that there are non-4-choosable planar graphs. Much attention has recently been put on sufficient conditions for planar graphs to be DP-$4$-colorable. In particular, for each $k \in \{3, 4, 5, 6\}$, every planar graph without $k$-cycles is DP-$4$-colorable. In this paper, we prove that every planar graph without $7$-cycles and butterflies is DP-$4$-colorable. Our proof can be easily modified to prove other sufficient conditions that forbid clusters …
A graph is {\em near-bipartite} if its vertex set can be partitioned into an independent set and a set that induces a forest. It is clear that near-bipartite graphs are $3$-colorable. In this note, we show that planar graphs without cycles of lengths in $\{4, 6, 8\}$ are near-bipartite.
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。
Planar Ramsey Numbers of Four Cycles Versus Wheels
对于两个给定的图$G$和$H$，平面Ramsey数$PR(G,H)$是最小整数$n$，使得$n$顶点上的每个可平面图$F$或者包含$G$的副本，或者其补图包含$H$的副本。本文首先刻画了无$C_4$平面图的一些结构性质，然后确定了无$C_4$平面图的所有平面Ramsey数$PR(C_4, W_n)$，对于$n\ge 3$ .
A graph $G$ is $(I,F)$-partitionable if its vertex set can be partitioned into two parts such that one part is an independent set, and the other induces a forest. In this paper, we prove that every planar graph without cycles of length $4, 6, 9$ is $(I,F)$-partitionable.
For large $n$ we determine exactly the maximum numbers of induced $C_4$ and $C_5$ subgraphs that a planar graph on $n$ vertices can contain. We show that $K_{2,n-2}$ uniquely achieves this maximum in the $C_4$ case, and we identify the graphs which achieve the maximum in the $C_5$ case. This extends work in a paper by Hakimi and Schmeichel and a paper by Ghosh, Győri, Janzer, Paulos, Salia, and Zamora which together determine both maxima asymptotically.
Improved Erdős-Pósa inequalities for odd cycles in planar graphs
本文改进了平面图中奇圈的埃罗什-波萨不等式，证明其比值不超过4，优于之前的6。
Min-max relations for odd cycles in planar graphs
设m（G）是图G的点不交奇圈的最大个数，t（G）是图G中使G为二分的顶点的最小个数.证明了当G是平面图时，t（G）<= 6 m（G）.这改进了Fiorini，Hardy，Reed和Vetta [Math. Program. B 110（2007），71-91]。
Planar graphs with distance of 3-cycles at least 2 and no cycles of lengths 5, 6, 7
本文证明3-圈距离至少为2且无5、6、7圈长的平面图是弱2-退化的，并可划分为一个无边子图和一个森林。
DP-coloring is a generalization of list coloring, which was introduced by Dvořák and Postle [J. Combin. Theory Ser. B 129 (2018) 38--54]. Zhang [Inform. Process. Lett. 113 (9) (2013) 354--356] showed that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles is 3-choosable. Liu et al. [Discrete Math. 342 (2019) 178--189] showed that every planar graph without 4-, 5-, 6- and 9-cycles is DP-3-colorable. In this paper, we show that every planar graph with neither adjacent triangles nor 5-, 6-, 9-cycles is DP-3-colorable, which generalizes these results. Yu et al. gave three Bord…
In this work we present an algorithm with which any arbitrary cubic planar map may be constructed through successive edge insertion while simultaneously constructing a set of proper edge labels and Hamiltonian cycles for each configuration. We present a publicly available Python implementation of this algorithm, and discuss both theoretical and numerical support for its validity, with reference to the well-known Four Colour Theorem.
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.
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.
In this paper we consider the number of Hamilton cycles in planar cubic graphs of high cyclic edge-connectivity, answering two questions raised by Chia and Thomassen ("On the number of longest and almost longest cycles in cubic graphs", Ars Combin., 104, 307--320, 2012) about extremal graphs in these families. In particular, we find families of cyclically $5$-edge connected planar cubic graphs with more Hamilton cycles than the generalized Petersen graphs $P(2n,2)$. The graphs themselves are fullerene graphs that correspond to certain carbon molecules known as nanotubes --- more precisely, the…
Orientable embeddings and orientable cycle double covers of projective-planar graphs
在图的闭2-胞腔嵌入中，每个面同胚于一个开圆盘，并且被图中的一个圈所包围。可定向强嵌入猜想是指每个2-连通图在某个可定向曲面中有一个闭的2-胞腔嵌入。这同时包含了圈双覆盖猜想和强嵌入猜想。本文证明了2-连通投影平面立方图在某个可定向曲面中有一个闭2-胞腔嵌入。证明的三个主要部分是：（1）将不可定向嵌入转化为可定向嵌入的外科手术方法;（2）三次图的可定向闭2-胞嵌入或可定向圈双覆盖的4-圈约化;（3）三次图的射影-平面嵌入的结构结果。我们证明了2-边连通的射影-可平面图（不一定是立方的）有可定向圈双覆盖。
Merker conjectured that if $k \ge 2$ is an integer and $G$ a 3-connected cubic planar graph of circumference at least $k$, then the set of cycle lengths of $G$ must contain at least one element of the interval $[k, 2k+2]$. We here prove that for every even integer $k \ge 6$ there is an infinite family of counterexamples.
$(3,1)^*$-choosability of planar graphs without adjacent short cycles
图$G$的列表分配是将颜色的列表$L(v)$分配给每个顶点$v\in V(G)$的函数$L$。$(L,d)^*$ -着色是将颜色$π(v)\in L(v)$分配给每个顶点$v\in V(G)$的映射$π$，使得至多$v$的$d$邻居接收颜色$π(v)$。一个图$G$称为$(k,d)^*$ -可选的，如果它允许对每个列表分配$L$都有一个$(L,d)^*$ -着色，而对所有的$v\in V(G)$都有$|L(v)|\ge k$. 2001年Lih等在\cite{LSWZ-01}中证明了不含4-圈和$l$ -圈的可平面图是$(3,1)^*$ -可选的，其中$l\in \{5,6,7\}$ .后来，Dong和Xu证明了不含4-圈和l-圈的可平面图是$(3,1)^*$ -可选的，其中$l\in \{8,9\}$ .存在含有4-圈的可平面图，它们不是$(3,1)^*$ -可选择的（Crown，Crown and Woodall，1986 \cite{CCW-86}）.这部分地解释了在所有上述已知的可平面图的$(3,1)^*$ -可选性的充分条件中4-圈是完全禁止的。在本文中，我们允许4-圈不相邻相对较短的圈。更准确地说，我们证明了每个平面图没有4-圈相邻的3-和4-圈是$(3,1)^*$ -选择。这是对上述所有成果的共同加强。作为结果，我们还部分地回答了Xu和Zhang的一个问题\cite{…
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 of Lam, Xu and Liu [Journal of Combin. Theory, Ser. B, 76 (1999) 117--126], of Cheng, Chen and Wang [Discrete Math., 339(2016) 3052--3057] and of Kim and Yu [arXiv:1709.09809v1].
Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers $d_0$ and $d_1$ such that planar graphs without $\{4,5\}$-cycles and $d^Δ\ge d_0$ are $3$-choosable and planar graphs without $\{4,5,6\}$-cycles and $d^Δ\ge d_1$ are $3$-choosable, where $d^Δ$ is the smallest distance between triangles. They showed that $2\le d_0\le 4$ and $d_1\le 3$. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without $\{4,5\}$-cycles and $d^Δ\ge 3$ are DP-$3$-colorable, and (2) planar graphs without $\{4,5,6\}$-cycles and $d^Δ\ge 2$ are DP-$3$-c…
In this paper we estimate the planar Turán number $\mathrm{ex}_\mathcal{P}(n,H)$ of some graphs $H$, i.e., the maximum number of edges in a planar graph $G$ of $n$ vertices not containing $H$ as a subgraph. We give a new, short proof when $H=C_5$, and study the cases when $G$ is bipartite or triangle-free and $H$ is a short even cycle. The proofs are mostly new applications or variants of the "contribution method" introduced by Ghosh, Győri, Martin, Paulos and Xiao in arXiv:2004.14094.
On Choosability with Separation of Planar Graphs with Forbidden Cycles
本文研究了图的列表着色的一种约束形式--带分离的可选性。图G上的一个（k，d）-列表分配L是一个函数，它给每个顶点v分配一个至少包含k种颜色的列表L（v），并且对于任意相邻的一对xy，列表L（x）和L（y）最多共享d种颜色。一个图G是（k，d）-可选的，如果对每个（k，d）-列表分配L，G都存在一个L-染色.这个概念也被称为分离的可选择性。证明了不含4-圈的可平面图是（3，1）-可选图，不含5-圈和6-圈的可平面图是（3，1）-可选图.此外，我们给出了另一个稍微强一点的证明，证明了无三角形可平面图是$(3,1)$ -可选的。
Planar Turán number of two adjacent cycles
该研究针对平面图Turán数问题，确定了由一条边连接两个不交环构成的$C_3$-$C_3$与$C_3$-$C_4$两类图的平面图Turán数，补充了相邻环结构的平面图极值边数结果。
List-coloring the Squares of Planar Graphs without 4-Cycles and 5-Cycles
设$G$是一个不含4-圈和5-圈的平面图，它具有最大度$Δ\ge 32$ .我们证明了$χ_{\ell}(G^2)\le Δ+3$ .对于任意大的最大度$Δ$，存在围长为6的可平面图$G_Δ$，且$χ(G_Δ^2)=Δ+2$ .因此，我们的界限在1以内是最优的。此外，我们的界限来自着色greetings在一个良好的秩序，所以立即扩展到在线列表着色的界限。此外，我们还证明了$L(p,q)$ -标记的界。具体地，$λ_{2,1}(G)\le Δ+8$，更一般地，$λ_{p,q}(G)\le (2q-1)Δ+6p-2q-2$，对于正整数$p$和$q$，具有$p\ge q$。同样，这些边界来自贪婪着色，所以它们立即扩展到这个问题的列表着色和在线列表着色变体。
Let $P(n,m)$ be a graph chosen uniformly at random from the class of all planar graphs on vertex set $\{1, \ldots, n\}$ with $m=m(n)$ edges. We study the cycle and block structure of $P(n,m)$ when $m\sim n/2$. More precisely, we determine the asymptotic order of the length of the longest and shortest cycle in $P(n,m)$ in the critical range when $m=n/2+o(n)$. In addition, we describe the block structure of $P(n,m)$ in the weakly supercritical regime when $n^{2/3}\ll m-n/2\ll n$.
The Minimum Number of $4$-Cycles in a Maximal Planar Graph with Small Number of Vertices
针对 Hakimi 与 Schmeichel 关于少量顶点极大平面图中4-环最小数目的未决猜想，本文予以证明，补全了该严格下界在所有小顶点数情形下的可达性。
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 (11:3)-colorable, a weakening of recently disproved Steinberg's conjecture. In particular, each such graph with n vertices has an independent set of size at least 3n/11.
The square of every subcubic planar graph without 4-cycles and 5-cycles is 7-choosable
本文证明了不含4-圈和5-圈的次三次平面图的平方是7-可选的，改进了Kim和Lian关于围长至少为6的次三次平面图的结果。
An improved integrality gap for disjoint cycles in planar graphs
提出新贪心取整算法解决平面图不可交叉环族环包装问题，将自然包装LP整数间隙上界改进至略小于3.5，并得出改进的Erdős-Pósa比率。
For a set of nonnegative integers $c_1, \ldots, c_k$, a $(c_1, c_2,\ldots, c_k)$-coloring of a graph $G$ is a partition of $V(G)$ into $V_1, \ldots, V_k$ such that for every $i$, $1\le i\le k, G[V_i]$ has maximum degree at most $c_i$. We prove that all planar graphs without 4-cycles and no less than two edges between triangles are $(2,0,0)$-colorable.
Planar graphs without 5-cycles and intersecting triangles are $(1,1,0)$-colorable
$G$的$(c_1,c_2,...,c_k)$ -着色是一个映射$φ:V(G)\mapsto\{1,2,...,k\}$，使得对于每个$i,1 \leq i \leq k$，$G[V_i]$的最大度至多为$c_i$，其中$G[V_i]$表示由着色$i$的顶点所诱导的子图. Borodin和Raspaud猜想：任何不含$5$ -圈和相交三角形的平面图都是$(0,0,0)$ -可着色的.本文证明了这类图是$(1,1,0)$ -可着色的.
Planar graphs having no cycle of length $4$, $6$ or $8$ are DP-3-colorable
本文证实了Dvořák和Postle的猜想，通过证明更强的结果：没有长度为4、6或8的圈的平面图是DP-3-可着色的。
1-Planar graphs without 6-cycles are 6-choosable
研究不含6-圈的1-平面图的可选择性问题，通过证明其为5-退化图，得出此类图是6-可选择的结论，核心方法是利用图的退化性及相关性质进行论证。
In 1976, Steinberg conjectured that planar graphs without $4$-cycles and $5$-cycles are $3$-colorable. This conjecture attracted numerous researchers for about 40 years, until it was recently disproved by Cohen-Addad et al. (2017). However, coloring planar graphs with restrictions on cycle lengths is still an active area of research, and the interest in this particular graph class remains. Let $G$ be a planar graph without $4$-cycles and $5$-cycles. For integers $d_1$ and $d_2$ satisfying $d_1+d_2\geq8$ and $d_2\geq d_1\geq 2$, it is known that $V(G)$ can be partitioned into two sets $V_1$ and…
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