All Questions
8 questions
13
votes
1
answer
933
views
Drawings of complete graphs with $Z(n)$ crossings
Hill conjectured that the minimum number of crossings in a drawing of the complete graph $K_n$ in the plane is exactly
$$Z(n) = \frac{1}{4} \bigg\lfloor\frac{n}{2}\bigg\rfloor \left\lfloor\frac{n-1}{...
- 6,101
9
votes
3
answers
470
views
Is it possible that every edge in a 1-planar drawing with minimum number of crossings is crossed?
A graph is 1-planar is it has drawing in the plane so that each edge is crossed at most once. Here we also assume the drawing satisfies (1) no edge is self-crossed; (2) no two adjacent edges are ...
- 557
2
votes
1
answer
127
views
The density of a tripartite 1-planar graph
1-planar graphs are those can be drawn in the plane so that there is at most one crossing per edge. We know that the maximum number of edges of an $n$-vertex 1-planar graph is at most $4n-8$, and the ...
- 1,190
1
vote
1
answer
194
views
Bounds on lengths of intervals in bounded-degree interval graphs
A graph is said to be an interval graph if its vertices can be associated with (closed) intervals on the real line $\mathbb R$ and there is an edge between two vertices if and only if the ...
- 277
1
vote
1
answer
98
views
Is there any known upper bound for the local crossing number of a graph drawing in the plane?
The local crossing number ${\rm LCR(G)}$ of a graph $G$ is defined as the least nonnegative integer $k$ such that the graph has a $k$-planar drawing. In other words, it is the smallest possible number ...
- 1,190
1
vote
1
answer
115
views
Bounds on lengths of boxes in bounded-degree box graphs
$\DeclareMathOperator{\box}{\operatorname{box}}$$\DeclareMathOperator{\cub}{\operatorname{cub}}$
This is a follow up and an extension of another question I asked recently.
A box graph is a graph ...
- 277
1
vote
1
answer
75
views
Given a vertex $u$ (of bounded degree $k$) and another vertex $v$ in a planar graph, what is the smallest number of "curves"?
Given a vertex $u$ (of bounded degree $k$) and another vertex $v$ in a planar graph $G$, what is the smallest number of "curves" in the plane drawn from $u$ to $v$ such that no $u$--$v$ path in $G$ ...
- 111
1
vote
0
answers
42
views
What lower bounds are known for pair crossing number and related questions in multigraphs?
So in terms of crossing number https://arxiv.org/pdf/1808.10480 gives a lower bound of $O(e^{2.5}/n^{1.5})$ for multigraphs with no face of length 2 with no node contained inside.
What do we know ...
- 111