Upper bounds for vertex degrees of planar $5$-chromatic graphs

Vertex Degrees in Planar Graphs

For a planar graph on n vertices we determine the maximum values for the following: 1) the sum of the m largest vertex degrees. 2) the number of vertices of degree at least k. 3) the sum of the degrees of vertices with degree at least k.

Vertex Degrees in Planar Maps

We prove a general multi-dimensional central limit theorem for the expected number of vertices of a given degree in the family of planar maps whose vertex degrees are restricted to an arbitrary (finite or infinite) set of positive integers D. We also discuss the possible extension to maps of higher genus.

Bounds for the Regularity of Edge Ideal of Vertex Decomposable and Shellable Graphs

Upper Bounds for Perfect Matchings in Pfaffian and Planar Graphs

We give upper bounds on weighted perfect matchings in Pfaffian graphs. These upper bounds are better than generalized Bregman’s inequality. We show that some of our upper bounds are sharp for 3 and 4-regular Pfaffian graphs. We apply our results to fullerene graphs.

New upper bounds on the decomposability of planar graphs

It is known that a planar graph on n vertices has branch-width/ tree-width bounded by ffiffiffi n p . In many algorithmic applications, it is useful to have a small bound on the constant . We give a proof of the best, so far, upper bound for the constant . In particular, for the case of tree-width, < 3:182 and for the case of branch-width, < 2:122. Our proof is based on the planar separation th...