17 research outputs found
The planar Tur\'an number of and
Let be a set of graphs. The planar Tur\'an number,
, is the maximum number of edges in an
-vertex planar graph which does not contain any member of as a
subgraph. When has only one element, we usually write
instead. The topic of extremal planar graphs was
initiated by Dowden (2016). He obtained sharp upper bound for both
and . Later on, we obtained
sharper bound for . In this paper, we give upper
bounds of and
. We also give constructions
which show the bounds are tight for infinitely many graphs.Comment: 11 pages, 11 figures. arXiv admin note: text overlap with
arXiv:2307.0690
On the maximum number of edges in planar graphs of bounded degree and matching number
We determine the maximum number of edges that a planar graph can have as a function of its maximum degree and matching number.publishedVersio
Dense circuit graphs and the planar Tur\'an number of a cycle
The of a
graph is the maximum number of edges in an -vertex planar graph without
as a subgraph. Let denote the cycle of length . The planar Tur\'an
number is known for . We show that
dense planar graphs with a certain connectivity property (known as circuit
graphs) contain large near triangulations, and we use this result to obtain
consequences for planar Tur\'an numbers. In particular, we prove that there is
a constant so that for all . When this bound is tight up to
the constant and proves a conjecture of Cranston, Lidick\'y, Liu, and
Shantanam