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An infinite family of octahedral crossing numbers
Author(s) -
Gross Jonathan L.
Publication year - 1978
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.3190020211
Subject(s) - combinatorics , mathematics , vertex (graph theory) , modulo , octahedron , graph , crossing number (knot theory) , discrete mathematics , crystallography , chemistry , intersection (aeronautics) , crystal structure , engineering , aerospace engineering
This paper calculates some crossing numbers for certain octahedral graphs. Precisely, if the number p is a prime power congruent to 1 modulo 4, then the crossing number of the p ‐dimensional octahedral graph in the orientable surface of genus ( p –1)( p ‐4)/4 is (p 2 –p)/2. The key step is the construction of a self‐dual imbedding of the complete graph on p vertices such that no face boundary contains a repeated vertex.