Last Car Decomposition of Planar Maps

The Last Temptation of William T. Tutte

In 1999, at one of his last public lectures, Tutte discussed a question he had considered since the times of the Four Color Conjecture. He asked whether the 4-coloring complex of a planar triangulation could have two components in which all colorings had the same parity. In this note we answer Tutte's question to the contrary of his speculations by showing that there are triangulations of the plane whose coloring complexes have arbitrarily many even and odd components. We end up with a closely related conjecture, which is based on an extensive computation, and which claims that for every plana…

面向树的外平面子图泛化平面图

Subgraph-universal planar graphs for trees

本文构造了顶点数约为 $O(n^{2.623})$ 的外平面图以包含所有 $n$ 顶点树作为子图，并确定了平面图包含三棵给定树所需的顶点数约为 $\frac{3}{2}n$，同时给出了包含所有（外）平面图的指数级下界。

Planarity of generalized ladder graphs

The Cartesian product of P_2 and P_n is called an n-ladder graph for a positive integer n. We call two paths P_m and P_n together with some edges each of which joins a vertex on P_m and a vertex on P_n a generalized (m,n)-ladder graph. In this paper, we completely characterize the planar generalized ladder graphs and the outerplanar generalized ladder graphs. A functigraph C(P_n ,f) is a generalized (n,n)-ladder graph. Consequently, our result solves the problem posed by A. Chen et al. (2011) to characterize planar functigraphs C(P_n ,f).

A new family of bijections for planar maps

We present bijections for the planar cases of two counting formulas on maps that arise from the KP hierarchy (Goulden-Jackson and Carrell-Chapuy formulas), relying on a "cut-and-slide" operation. This is the first time a bijective proof is given for quadratic map-counting formulas derived from the KP hierarchy. Up to now, only the linear one-faced case was known (Harer-Zagier recurrence and Chapuy-Féray-Fusy bijection). As far as we know, this bijection is new and not equivalent to any of the well-known bijections between planar maps and tree-like objects.

Tree-decorated planar maps

We introduce the set of (non-spanning) tree-decorated planar maps, and show that they are in bijection with the Cartesian product between the set of trees and the set of maps with a simple boundary. As a consequence, we count the number of tree decorated triangulations and quadrangulations with a given amount of faces and for a given size of the tree. Finally, we generalise the bijection to study other types of decorated planar maps and obtain explicit counting formulas for them.

Coloring outerplanar graphs and planar 3-trees with small monochromatic components

In this work, we continue the study of vertex colorings of graphs, in which adjacent vertices are allowed to be of the same color as long as each monochromatic connected component is of relatively small cardinality. We focus on colorings with two and three available colors and present improved bounds on the size of the monochromatic connected components for two meaningful subclasses of planar graphs, namely maximal outerplanar graphs and complete planar 3-trees.