
Structure and generation of crossingcritical graphs
We study ccrossingcritical graphs, which are the minimal graphs that r...
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Tight Upper Bounds on the Crossing Number in a MinorClosed Class
The crossing number of a graph is the minimum number of crossings in a d...
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Meyniel's conjecture on graphs of bounded degree
The game of Cops and Robbers is a well known pursuitevasion game played...
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Crossing Numbers and Stress of Random Graphs
Consider a random geometric graph over a random point process in R^d. Tw...
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Tuza's Conjecture for Threshold Graphs
Tuza famously conjectured in 1981 that in a graph without k+1 edgedisjo...
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A local epsilon version of Reed's Conjecture
In 1998, Reed conjectured that every graph G satisfies χ(G) ≤1/2(Δ(G) + ...
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The complexity of approximating the matching polynomial in the complex plane
We study the problem of approximating the value of the matching polynomi...
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Bounded maximum degree conjecture holds precisely for ccrossingcritical graphs with c ≤ 12
We study ccrossingcritical graphs, which are the minimal graphs that require at least c edgecrossings when drawn in the plane. For every fixed pair of integers with c> 13 and d> 1, we give first explicit constructions of ccrossingcritical graphs containing a vertex of degree greater than d. We also show that such unbounded degree constructions do not exist for c< 12, precisely, that there exists a constant D such that every ccrossingcritical graph with c< 12 has maximum degree at most D. Hence, the bounded maximum degree conjecture of ccrossingcritical graphs, which was generally disproved in 2010 by Dvořák and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values c< 12.
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