Relationships between faces in regular, connected and planar graphs

Abstract

This study focuses on estimating the predicted ultimate load of the pile that does not reach the failure, based on five methods: Chin-kondner Extrapolation, DeBeer, Shen,  which estimates the value of the failure load of the pile, and by using the numerical methods. This paper compares the load results obtained from the field for the North Nasiriyah Bridge project and the results obtained from numerical methods. Plaxis3d Foundation 2020 was adopted to clarify the extent to which numerical methods can approach the behavior of the actual pile and give results that are close to the field. Four bored piles (TP1, TP2, TP3, TP4) were used in this paper, and their testing load was 60, 650, 950, 70 tons, respectively. In addition, the Mohr-Coulomb model, which considers the site's soil parameters, was employed to simulate both the mound and the soil Let G a planar graph, connected and 3-regular, if the whole face, which G contains, is triangular or hexagonal then the number of triangles in it is exactly four. This truth is known from [4].

Two extensions to the previous fact are the purpose of this paper:

The first, to show, assuming that given a graph G that is planar, connected and s-regular, if G contains two types of faces, one of them contains double the number of edges than the other, (suppose l and 2l), and if    (or  ) then the number of the faces which contains less edges, is always four.

The second, to show, assuming that given a graph G that is planar, connected and s-regular, if G contains two types of faces, one of them contains, multiple number of edges than the other, (suppose l and jl), and if   then the number of the faces which contains less edges, is always three or four.