3-trees with few vertices of degree 3 in circuit graphs |
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Authors: | Atsuhiro Nakamoto Yoshiaki Oda |
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Institution: | a Department of Mathematics, Faculty of Education and Human Sciences, Yokohama National University, 79-2 Tokiwadai, Hodogaya-ku, Yokohama 240-8501, Japan b Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan |
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Abstract: | A circuit graph(G,C) is a 2-connected plane graph G with an outer cycle C such that from each inner vertex v, there are three disjoint paths to C. In this paper, we shall show that a circuit graph with n vertices has a 3-tree (i.e., a spanning tree with maximum degree at most 3) with at most vertices of degree 3. Our estimation for the number of vertices of degree 3 is sharp. Using this result, we prove that a 3-connected graph with n vertices on a surface Fχ with Euler characteristic χ≥0 has a 3-tree with at most vertices of degree 3, where cχ is a constant depending only on Fχ. |
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Keywords: | 3-connected graph Circuit graph 3-tree Surface |
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