Drawing disconnected graphs on surfaces

Importance: Medium ✭✭
Recomm. for undergrads: no
Posted by: mdevos
on: May 12th, 2007

\begin{conjecture} Let $G$ be the disjoint union of the graphs $G_1$ and $G_2$ and let $\Sigma$ be a surface. Is it true that every optimal drawing of $G$ on $\Sigma$ has the property that $G_1$ and $G_2$ are disjoint? \end{conjecture}

We insist on the usual restrictions for drawings (as in \OPref[The Crossing Number of the Complete Graph]{the_crossing_number_of_the_complete_graph}).

Although both crossing numbers and embeddings of graphs on general surfaces are rich and well-studied subjects, their common generalization - drawing graphs on general surfaces has received very little attention. The question highlighted here appears to be quite basic in nature, but due to the combined difficulties of crossings and general surfaces, it may be quite difficult to resolve.

This conjecture is trivially true when $\Sigma$ is the plane, and DeVos, Mohar, and Samal have proved that it also holds when $\Sigma$ is the projective plane. It is open for all other surfaces to the best of my (M. DeVos) knowledge.


% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.

Drawing disconnected graphs on surfaces : any reference ?


You don't mention reference in this problem, though it is said that some work has been made. Would it be possible to know how the projective plane has been shown to verify this conjecture ?

Thanks in advance for any piece of information.

Laurent Beaudou

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