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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 ...
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 ...
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 ...
1 vote
0 answers
97 views

Problem related to crossing number

Let $G$ be a graph embedded in the plane (with crossings). For $ F \subset E(G) $, denote by $c(F)$ the set of edges of $G$ that cross some edge in $F$. Denote $\delta(v)$ the set of edges with one ...
4 votes
1 answer
214 views

Find all 2-planar drawings of $K_6$ and $K_7$

A $k$-planar graph is a graph which can be embedded with at most $k$ crossings per edge. It is proved that a complete graph $K_n$ is 2-planar if and only if $n\le 7$. Angelini P., Bekos M. A., ...
4 votes
1 answer
393 views

Example to show pairwise crossing number is not equal to crossing number

A common point of two edges in a graph drawing that is not an incident vertex is called a crossing. The crossing number $cr(G)$ is defined to be the minimum number of crossings in any drawing of $G$....
5 votes
1 answer
479 views

Can all crossings in a graph be moved to one point?

Consider a graph $G$ with at least two unavoidable crossings, say, the disjoint union of two copies of $K_5$. Can such a graph always be drawn so that there is only one singular point (where all ...