Download PDF by Claude Berge (auth.), Siddani Bhaskara Rao (eds.): Combinatorics and Graph Theory: Proceedings of the Symposium

By Claude Berge (auth.), Siddani Bhaskara Rao (eds.)

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Extra resources for Combinatorics and Graph Theory: Proceedings of the Symposium Held at the Indian Statistical Institute, Calcutta, February 25–29, 1980

Example text

14) is just a homogenizing variable. (r-s)! Sr, s (A) x n-r X s p r-s ... (15) r=O s=0 This follows by an obvious substitution in (14). 's lJ and S r,s (6) give the interrelations between 's. Thus, from (14) we have n i n-i xj . n r r,s(X_p)n-r(~_p)s ~ c x pz-j = ~ ~ (r-s)~ S r=O s=O i=0 j=O uj -- nr z__ z (r-s): (nrn~r) s r 0 s=O r,s P r-s x1(_p)n r i ) i=O { j ! O ( ~ ) x j ( _ p ) s - j pr-s> Therefore en_i, j = n-i r (n-r) ~ (_l)n-i-j-(r-s)s ~ Z (r-s)! ( ) r=O s=j i r,s or, el, j = i (-i) i-j 2 r=O r ~ ( r - s ) .

If i P2's The status of the converse of this theorem is not very clear. 9. CONCLUSION We have introduced five different graph polynomials the P,Q,T,R and nomials and conjectured that any of these is a complete graph invariant. blished the equivalence of the first three in Theorems 1 and 4. the implications F + R + T in the sense that The converse implications are not clear. R F poly- We have esta- We have also establishec is derivable from F and T from R. Thus, as things stand, the permanent conjecture (the first invariant conjecture) may be false even if the frame conjecture is true.

B 69(1965), 125-130. 2. J. Edmonds, 3. J. Edmonds, Submodular functions, matroids, and certain polyhedra, in : Combinatorial Structures and their AppI. (Proc. Intern. Conf. Calgary, 1969; R. Guy, H. Hanani, N. Sauer, J. Y. 1970, 69-87. Optimum branchings, J. Res Nat. Bur. Stan. B 71(1967), 233-240. 4. J. Edmonds, Edge-disjoint branchings, in : Combinatorial Algorithms (Courant Comp. Sci. Symp. Monterey, 1972; R. Y. 1973, 91-96. 5. J. Edmonds, 6. J. Edmonds and R. , i(1977), 185-204. 7. J. L. Johnson, Matching, Euler tours, and the Chinese postman, Math.

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