And it’s even harder to tell whether a complicated planar graph stays planar when you start adding edges as you might when planning a new stretch of highway. Computer scientists have been searching ...

Graph colouring remains a central topic in graph theory, providing the mathematical framework for assigning colours to the elements of a graph under specific constraints. In particular, the ...

The term "minimal nonplanar subgraph" of a graph G refers to a nonplanar subgraph N of G having the property that each proper subgraph of N is planar. A well-known result of graph theory is that there ...

Let us say that a graph is k- apex if it contains a set of at most k vertices whose removal yields a planar graph. We define the apex number of a graph G as the minimum k for which G is k-apex.

We prove that any distributional limit of finite planar graphs in which the degree of the root has an exponential tail is almost surely recurrent. As a corollary, we obtain that the uniform infinite ...