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Székely László: Drawings and Crossings in Infinite Graphs |
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| Much attention has been paid to infinite planar graphs. We investigate the ordinary crossing numbers of infinite graphs. Building heavily on the theory of infinite planar graphs, and slightly extending it to multigraphs, we study graph embeddings, graph drawings, and crossing numbers of infinite graphs. We investigate what kind of properties can be forced on optimal drawings, including the avoidance of certain kinds of accumulation points which can emerge in drawings of infinite graphs. Among other results, we extend the equality of planar and spherical crossing numbers to infinite graphs, classify outerplanar graphs, and prove compactness results for the crossing number of countable graphs and the $k$-page crossing numbers of countable graphs. A sample result is that on the 2-page book, continuum many even cycles can be drawn without crossing, but not uncountably many odd cycles. Joint work with Eva Czabarka and Alec Helm. |
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