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@article{1600841, author = {Hliněný, Petr and Korbela, Michal}, article_number = {3}, keywords = {graph; crossing number; crossing-critical}, language = {eng}, issn = {0231-6986}, journal = {Acta Math. Univ. Comenianae}, title = {On the achievable average degrees in 2-crossing-critical graphs}, url = {http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1178/725}, volume = {88}, year = {2019} }
TY - JOUR ID - 1600841 AU - Hliněný, Petr - Korbela, Michal PY - 2019 TI - On the achievable average degrees in 2-crossing-critical graphs JF - Acta Math. Univ. Comenianae VL - 88 IS - 3 SP - 787-793 EP - 787-793 SN - 02316986 KW - graph KW - crossing number KW - crossing-critical UR - http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1178/725 L2 - http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1178/725 N2 - c-Crossing-critical graphs are the minimal graphs requiring at least c edge crossings in every drawing in the plane. The structure of these obstructions is very rich for every c>=2. Although, at least in the first nontrivial case of c=2, their structure is well understood. For example, we know that, aside of finitely many small exceptions, the 2-crossing-critical graphs have vertex degrees from the set {3, 4, 5, 6} and their average degree can achieve exactly all rational values from the interval [3+1/2 , 4+2/3]. Continuing in depth in this research direction, we determine which average degrees of 2-crossing-critical graphs are possible if we restrict their vertex degrees to proper subsets of {3, 4, 5, 6}. In particular, we identify the (surprising) subcases in which, by number-theoretical reasons, the achievable average degrees form discontinuous sets of rationals. ER -
HLINĚNÝ, Petr a Michal KORBELA. On the achievable average degrees in 2-crossing-critical graphs. \textit{Acta Math. Univ. Comenianae}. 2019, roč.~88, č.~3, s.~787-793. ISSN~0231-6986.
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