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Note on the Pair-crossing Number and the Odd-crossing Number.

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Title: Note on the Pair-crossing Number and the Odd-crossing Number.
Authors: Géza Tóth1
Source: Discrete & Computational Geometry. Jun2008, Vol. 39 Issue 4, p791-799. 9p. 2 Diagrams, 2 Graphs, 1 Map.
Subjects: Mathematics, Writing of numerals, Roman numerals, Numerals
Abstract:
Abstract   The crossing number ${\mbox{\sc cr}}(G)$ of a graph G is the minimum possible number of edge-crossings in a drawing of G, the pair-crossing number ${\mbox{\sc pair-cr}}(G)$ is the minimum possible number of crossing pairs of edges in a drawing of G, and the odd-crossing number ${\mbox{\sc odd-cr}}(G)$ is the minimum number of pairs of edges that cross an odd number of times. Clearly, ${\mbox{\sc odd-cr}}(G)\le {\mbox{\sc pair-cr}}(G)\le {\mbox{\sc cr}}(G)$ . We construct graphs with $0.855\cdot {\mbox{\sc pair-cr}}(G)\ge {\mbox{\sc odd-cr}}(G)$ . This improves the bound of Pelsmajer, Schaefer and Štefankovič. Our construction also answers an old question of Tutte.
Slightly improving the bound of Valtr, we also show that if the pair-crossing number of G is k, then its crossing number is at most O(k 2/log 2 k).
[ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
Abstract:<div class="Abstract"><a name="Abs1"></a><span class="AbstractHeading">Abstract  </span> The crossing number <a name="IEq1"></a><img src="/fulltext-image.asp?format=htmlnonpaginated&src=2M236023N5615716_html/454_2007_9024_Article_IEq1.gif" alt="${\mbox{\sc cr}}(G)$" align="middle" border="0"> of a graph G is the minimum possible number of edge-crossings in a drawing of G, the pair-crossing number <a name="IEq2"></a><img src="/fulltext-image.asp?format=htmlnonpaginated&src=2M236023N5615716_html/454_2007_9024_Article_IEq2.gif" alt="${\mbox{\sc pair-cr}}(G)$" align="middle" border="0"> is the minimum possible number of crossing pairs of edges in a drawing of G, and the odd-crossing number <a name="IEq3"></a><img src="/fulltext-image.asp?format=htmlnonpaginated&src=2M236023N5615716_html/454_2007_9024_Article_IEq3.gif" alt="${\mbox{\sc odd-cr}}(G)$" align="middle" border="0"> is the minimum number of pairs of edges that cross an odd number of times. Clearly, <a name="IEq4"></a><img src="/fulltext-image.asp?format=htmlnonpaginated&src=2M236023N5615716_html/454_2007_9024_Article_IEq4.gif" alt="${\mbox{\sc odd-cr}}(G)\le {\mbox{\sc pair-cr}}(G)\le {\mbox{\sc cr}}(G)$" align="middle" border="0"> . We construct graphs with <a name="IEq5"></a><img src="/fulltext-image.asp?format=htmlnonpaginated&src=2M236023N5615716_html/454_2007_9024_Article_IEq5.gif" alt="$0.855\cdot {\mbox{\sc pair-cr}}(G)\ge {\mbox{\sc odd-cr}}(G)$" align="middle" border="0"> . This improves the bound of Pelsmajer, Schaefer and Štefankovič. Our construction also answers an old question of Tutte. <div class="AbstractPara"> <div class=""> Slightly improving the bound of Valtr, we also show that if the pair-crossing number of G is k, then its crossing number is at most O(k 2/log 2 k). </div> </div> </div> [ABSTRACT FROM AUTHOR]
ISSN:01795376