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Extra info for Plane trigonometry, by S.L. Loney.
We remark that such paths can be defined for any c o n n e c t e d graph e m b e d d e d on any o r i e n t a b l e surface; it is easy to see that they are in- trinsic to such an embedded graph, and that in the ease of a 3-connected planar graph they are intrinsic to the graph itself. The neighbours of an edge is leftmost or rightmost. e. e are the (-< 4) edges for which Figure 4 shows an edge e e and 4 neighbours • i Suppose the family of left-right paths that belongs to Pl, P2, once in "''' Pk" 3.
G. Cerf, D. D. Cowan, R. C. Mullin, Networks and Generalized Series R. G. Stanton, Moore Graphs, and (April, 1974). Computer Congressus Numerantium 9, Proc. Third Manitoba Conference on Numerical Mathematics (1973), 379-398. Graph Theory (Addison-Wesley,  F. Harary,  A. J. Hoffman and R. R. , 1969) IBM Journal of Research and Development (1960), 497-504. A. Department of Computer University of Waterloo Science Ontario Canada Department of Combinatorics University of Waterloo and Optimization Ontario Canada Department of Computer University of Manitoba Science and Faculty of Mathematics University of Newcastle Winnipeg New South Wales Canada Australia A NOTE ON BALANCED WEIGHING MATRICES R, C, MULLIN A balanced w e i g h i n g m a t r i x is a square o r t h o g o n a l m a t r i x of l's and -l's O's, such that the m a t r i x obtained by squaring entries is the incidence m a t r i x of a (v, k, l) configuration.
With a desired dimension of that H. For instance, of Z(G) This it n B(G) 51 . ili~ili' ~ :::::" }° Figure 9 Figure 10 Figure 11 ... 52 i < Figure 12 53 by a process that might be d e s c r i b e d as the s u p e r p o s i t i o n of squiggles. The graphs all have G c o r r e s p o n d i n g to the 4-regular graphs H of Figure 12 dim Z n B = 2. Corollary: an arbitrary Let dim Z n B = k > 0. Then sequence of graph having an even number of spanning Corollary. sequences A family k edge contractions k-1 trees.