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An inequality chain of domination parameters for trees

E.J. Cockayne, O. Favaron, J. Puech, C.M. Mynhardt (1998)

Discussiones Mathematicae Graph Theory

We prove that the smallest cardinality of a maximal packing in any tree is at most the cardinality of an R-annihilated set. As a corollary to this result we point out that a set of parameters of trees involving packing, perfect neighbourhood, R-annihilated, irredundant and dominating sets is totally ordered. The class of trees for which all these parameters are equal is described and we give an example of a tree in which most of them are distinct.

An inequality concerning edges of minor weight in convex 3-polytopes

Igor Fabrici, Stanislav Jendrol' (1996)

Discussiones Mathematicae Graph Theory

Let e i j be the number of edges in a convex 3-polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is 20 e 3 , 3 + 25 e 3 , 4 + 16 e 3 , 5 + 10 e 3 , 6 + 6 [ 2 / 3 ] e 3 , 7 + 5 e 3 , 8 + 2 [ 1 / 2 ] e 3 , 9 + 2 e 3 , 10 + 16 [ 2 / 3 ] e 4 , 4 + 11 e 4 , 5 + 5 e 4 , 6 + 1 [ 2 / 3 ] e 4 , 7 + 5 [ 1 / 3 ] e 5 , 5 + 2 e 5 , 6 ≥ 120 ; moreover, each coefficient is the best possible. This result brings a final answer to the conjecture raised by B. Grünbaum in 1973.

An introduction to finite fibonomial calculus

Ewa Krot (2004)

Open Mathematics

This is an indicatory presentation of main definitions and theorems of Fibonomial Calculus which is a special case of ψ-extented Rota's finite operator calculus [7].

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