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 cycle of length at least $\frac{2n+4}{5}$, and this result has recently been improved multiple times.
In this paper, we prove that every essentially 4-connected planar graph $G$ on $n$ vertices contains a cycle of length at least $\frac{5}{8}(n+2)$. This improves the previously best-known lower bound $\frac{3}{5}(n+2)$.
A note on cycles in cyclically $4$-edge-connected cubic planar graphs
School of Mathematical Sciences, Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education, Tongji University（同济大学数学科学学院，智能计算与应用教育部重点实验室，同济大学）
DP-3-coloring of planar graphs without $4,9$-cycles and two cycles from $\{5,6,7,8\}$
列表着色的一个推广，现在称为DP着色，最近由Dvopelák和Postle引入。本质上，DP着色在邻顶点的颜色列表之间分配任意匹配，而不是像列表着色那样仅匹配相同的颜色。关于可平面图列表染色的一些结果已经被推广到DP染色的情形。我们注意到列表着色结果并不总是扩展到DP着色结果。本文的主要结果是证明了每个不含长为$\{4, a, b, 9\}$ for $a, b \in \{6, 7, 8\}$的圈的平面图都是DP-$3$-可着色的，推广了关于可平面图的$3$ -可选性的三个已有结果.
The role of planarity in connectivity problems parameterized by treewidth
For some years it was believed that for "connectivity" problems such as Hamiltonian Cycle, algorithms running in time 2^{O(tw)}n^{O(1)} -called single-exponential- existed only on planar and other sparse graph classes, where tw stands for the treewidth of the n-vertex input graph. This was recently disproved by Cygan et al. [FOCS 2011], Bodlaender et al. [ICALP 2013], and Fomin et al. [SODA 2014], who provided single-exponential algorithms on general graphs for essentially all connectivity problems that were known to be solvable in single-exponential time on sparse graphs. In this article we f…
考虑以下两种方法来给图的顶点着色，其中放松了邻顶点获得不同颜色的要求。如果每个单色分量的最大度数至多为$d$，则着色具有“缺陷”$d$。如果每个单色分量最多有$c$顶点，则着色具有“聚类”$c$。本文调查研究这些类型的着色，其中的第一要务是尽量减少颜色的数量，小缺陷或小集群作为次要目标。列表着色变体也被考虑。研究了以下图形类：外可平面图、可平面图、可嵌入曲面中的图、具有给定最大度的图、具有给定最大平均度的图、不包括给定子图的图、具有线性交叉数的图、无链接或无链接可嵌入图、具有给定Colin de Verdière参数的图、具有给定周长的图、不包括作为浸入的固定图的图、具有给定厚度的图、具有给定堆叠数或堆叠数的图、不包括$K_t$作为子图的图、不包括$K_{s,t}$作为子图的图、以及不包括任意图$H$作为子图的图。几个开放的问题进行了讨论。
