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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Additional resources for Combinatorial Mathematics III: Proceedings of the Third Australian Conference Held at the University of Queensland, 16–18 May, 1974
Corollary. N(G), the number of spanning trees of has just one left-right path. T h e o r e m 4. and only if G Proof. u(G) is the set of edges having cycle character. Hz(E) (mod 2). It is w e l l - k n o w n that N(G) = N(G-e) + N(G e) u(G) G so that ~ N(G) single spanning tree. case is odd if Using the n o t a t i o n of Lemma 5, we want to show that z N(G) show that G, when u(G) Corollary. satisfies G is "small", that is, w h e n G has a We leave to the reader the v e r i f i c a t i o n that in is a tree, with, in the plane, N(G) (using Lemma 5) it is sufficient to possibly, loops a t t a c h e d at vertices, embedded = 1.
The theorem and c o n s i d e r is true the case whenever ~ = q.
A! ~) "- .... _.. -i" Ii A /\ ,---~--~ ~-~-~- -- Figure 1 Such a path is c o n t i n u e d fo~ one period. Accordingly, on such a path, each edge t r a v e r s e d will have been traversed either once right) or twice (left and right). (left or It is clear that all edges of G can be covered by a family of one or more left-right paths such that each edge occurs exactly twice (once left, once right) on paths of the family. The d i r e c t i o n of any path is irrelevant. However, if an edge is traversed twice by the same path, whether in the same or in opposite directions is important.
Combinatorial Mathematics III: Proceedings of the Third Australian Conference Held at the University of Queensland, 16–18 May, 1974 by V. G. Cerf, D. D. Cowan, R. C. Mullin, R. G. Stanton (auth.), Dr. Anne Penfold Street, Dr. Walter Denis Wallis (eds.)