2013/05/02 by Matthew Babbitt, Babbitt, Matthew, J. T. Geneson +3
Computer Science · Mathematics · #05C35 #Combinatorics (math.CO) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #cs.DM #math.CO #msc:05C35
paper · pdf · doi:10.48550/arxiv.1305.0505
17 pages, 6 figures
arxiv created 2014/11/13 · arxiv updated 2014/11/14
We examine several types of visibility graphs in which sightlines can pass through k objects. For k ≥ 1 we bound the maximum thickness of semi-bar k-visibility graphs between \lceil (2)/(3) (k + 1) \rceil and 2k. In addition we show that the maximum number of edges in arc and circle k-visibility graphs on n vertices is at most (k+1)(3n-k-2) for n > 4k+4 and n \choose 2 for n ≤ 4k+4, while the maximum chromatic number is at most 6k+6. In semi-arc k-visibility graphs on n vertices, we show that the maximum number of edges is n \choose 2 for n ≤ 3k+3 and at most (k+1)(2n-(k+2)/(2)) for n > 3k+3, while the maximum chromatic number is at most 4k+4.