By V. G. Cerf, D. D. Cowan, R. C. Mullin, R. G. Stanton (auth.), Dr. Anne Penfold Street, Dr. Walter Denis Wallis (eds.)
Read or Download Combinatorial Mathematics III: Proceedings of the Third Australian Conference Held at the University of Queensland, 16–18 May, 1974 PDF
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Extra resources for Combinatorial Mathematics III: Proceedings of the Third Australian Conference Held at the University of Queensland, 16–18 May, 1974
It appears difficult to obtain a s a t i s f a c t o r y characterization. It has been shown by J. Horton  that if a cyclic sequence of edges in w h i d h each edge appears twice is that e n c o u n t e r e d on a left-right path of plane graph Corollary. ) be the number of paths of a 4-regular plane that always violate trated in Figure is unique the component-defining requirement 10, and that contain each edge exactly is odd if and only if H once. illus- Then k has exactly one component. (The r e a d e r is r e f e r r e d to [S] for a c o r r e s p o n d e n c e between such paths and spanning trees of Corollary.
N = 28) are H a m i l t o n i a n However, is a n o n - H a m i l t o n i a n example for the M(28,3). 23 I Figure 2 26A 14 (from 18) 8 12 3, 10 i\ l 5 Fisure 11 26B (from 20A) 24 14 10 8 ~4 2 Figure 26C 12 (from 20B) 11 6 5 1 13 Figure 14 26D (from 22A) v Do o~ ~rj v t~ r~ M o~ ° ~ 26 13 14 10 /1 2 Figure 1 26G (from 23C) 10 13 12 7 1 Figure 26H (from 25A) 27 REFERENCES LI] V. G. Cerf, D. D. Cowan, R. C. Mullin, Topological R. G. Stanton, Design Considerations in Computer Communications Networks in Computer Communication Networks, ed.
2. N E C E S S A R Y CONDITIONS FOR THE EXISTENCE OF BALANCED W E I G H I N G MATRICES Since a BW(v,k) implies the existence of a (v,k,l) configuration, the following conditions are known to be necessary: (i) if v is even then (k-l) must be a perfect square; 29 (ii) if v is odd then the equation x 2 = (k-l)y 2 + (-1) (v-1)/2 It is also have (See [7, trivial (iii) Further must a solution that it is s h o w n for a in for a W(v,k) k (v) if v is odd, then (v-k) 2 - (v-k) shown parameters p It was shown any p r o j e c t i v e of a w e i g h i n g incidence matrix in PG(m,n) where matrix of the signed if b o t h m and construction it is shown a It is n o t e d matrices true part incidence k must is s t r o n g e r is a s o l u t i o n for that k-l is not of of not than (1); (i) but implies matrix k (ii) the is has a divisible of the to as certain signing).