On 3-Connected Plane Graphs without Triangular Faces

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Abstract

We prove that each polyhedral triangular face free map G on a compact 2-dimensional manifold M with Euler characteristic χ(M) contains a k-path, i.e., a path on k vertices, such that each vertex of this path has, in G, degree at most (5/2)k if M is a sphere S0 and at most (k/2)⌊(5+49-24χ(M))/2⌋ if M≠S0 or does not contain any k-path. We show that for even k this bound is best possible. Moreover, we show that for any graph other than a path no similar estimation exists. © 1999 Academic Press.

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Harant, J., Jendrol’, S., & Tkác, M. (1999). On 3-Connected Plane Graphs without Triangular Faces. Journal of Combinatorial Theory. Series B, 77(1), 150–161. https://doi.org/10.1006/jctb.1999.1918

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