Abstract
We prove that there are intersection graphs of axis-aligned boxes in R3 and intersection graphs of straight lines in R3 that have arbitrarily large girth and chromatic number.
Highlights
Erdos [8] proved that there exist graphs with arbitrarily large girth and chromatic number
We prove that there exist such graphs that can be realised geometrically as intersection graphs of axis-aligned boxes in R3 and of straight lines in R3
In light of Burling’s construction, the problem of whether intersection graphs of axis-aligned boxes in R3 with large girth have bounded chromatic number was raised by Kostochka and Perepelitsa [17]
Summary
Erdos [8] proved that there exist graphs with arbitrarily large girth and chromatic number. There are intersection graphs of axis-aligned boxes in R3 with arbitrarily large girth and chromatic number. There are intersection graphs of straight lines in R3 with arbitrarily large girth and chromatic number.
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