The first part of this paper deals with the properties of C 3 -saturated graphs. It will be shown that for any C 3 -saturated graph, G,with which two vertices are adjacent iff the distance between them, in G, is k. In addition to this, a full description of the set of planar C 3 -saturated graphs, PSAT(n, C 3 ), will be given. It will be shown that there are only three kinds of such graphs. In the second part of the paper a useful characterization of graphs which are C 3 -saturated and C 4 -free will be given in terms of the adjacency and incidence matrices A and B.

An image of a plane graph, G = (V, E) of order n and size m, is said to be an edge-magic plane graph if there is a bijection f : E → {1, 2, .., m} such that for all s−side faces of G, except the infinite face, the sum of the labels of its edges is a constant k(s). Such a bijection will be called an edge-magic plane labeling of G. In case that all the finite sides of a graph G having the same size we will be interested in determining the minimum and the maximum number, k, such that there exists an edge-magic plane labeling of G, in which k is the sum of the edge labeling of each face. In this paper we find such a minimum and maximum numbers for a wheel with even order. Furthermore we conjecture that the same formula is valid for the odd case.

An image of a plane graph, G = (V, E) of order n and size m, is said to be a vertex-edge-magic plane graph if there is a bijection f : V ∪ E → {1, 2, .., n + m} such that for all s − side faces of G, except the infinite face, the sum of the labels of its vertices and edges is a constant k(s). Such a bijection will be called a vertex-edge-magic plane labeling of G. In case that all the finite sides of a graph G having the same size we will be interested in determining the minimum and the maximum number, k, such that there exists a vertex-edgemagic labeling of G, in which k is the sum of the vertex and edge labeling of each face. In this paper we find such a minimum and maximum numbers for a wheel with even order.

B. Erubin 43b describes a special tube of Rabban Gamaliel, with which he was able to measure distances of up to two thousand cubits on a plane, and which he would also use to measure the depth of ravines. With this tube, he could also measure angles, or at least set the tube on a specific angle to measure distances using congruent triangles. Although the method of measurement presented in the Talmud is not clear, very few sages thoroughly studied or interpreted the measurement method. Some have understood the method, on the basis of their own contemporary mathematical knowledge, while others simply laconically quoted their predecessors without understanding the earlier sages’ explanations. Notably, in Rabban Gamaliel’s period, optical lenses were not in use. Instead, a hollow tube was adapted to allow the measurement of a fixed distance. There are diverse opinions on how this tube was used to measure distance, depth, and height. In this paper we address the measurement methods of the Geonic sages and their subsequent interpreters and assess the methods they propose.

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