The height of a 4-cycle in triangle-free 1-planar graphs with minimum degree 5 |
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Authors: | O V Borodin I G Dmitriev A O Ivanova |
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Institution: | (1) Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia;(2) Yakutsk State University, ul. Kulakovskogo 48, Yakutsk, 677000, Russia |
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Abstract: | A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. It is known that each 1-planar graph has a vertex of degree at most 7, and also either a vertex of degree at most 4 or a cycle of length at most 4. In the article, it is proven that each triangle-free 1-planar graph of degree less than 5 has a 4-cycle that consists of vertices of degree at most 8. |
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