It is shown that given a sequence of integers =< d1; d2; : : : ; dn >, it is possible in O(n) time to determine whether there exists a 2-tree that realizes the degree sequence , and if so, to construct one.

We show that the Erdos-Gallai condition characterizing graphical degree sequences of length p needs to be checked only for as many n as there are distinct terms in the sequence, not all n, 1 6 n 6 p. MSC: 05 C07

We prove that the existence of a homogeneous invariant of degree n for a representation of a semi-simple Lie group guarantees the existence of non-trivial solutions of D α = 0: these correspond to the maximum value of the square of the invariant divided by the norm of the representation to the n th power.

A sequence S is potentially K4 − e graphical if it has a realization containing a K4 − e as a subgraph. Let σ(K4 − e, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(K4 − e, n) is potentially K4 − e graphical. Gould, Jacobson, Lehel raised the problem of determining the value of σ(K4 − e, n). In this paper, we prove that σ(K4 − e, n) = 2[(3n− 1)/2] fo...

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph G with ( n+1 2 ) edges admits an edge decomposition G = H1 ⊕ · · · ⊕Hn such that Hi has i edges and is isomorphic to a subgraph of Hi+1, i = 1, . . . , n−1. We show that every bipartite graph G with ( n+1 2 ) edges such that the degree sequence d1, . . . , dk of one of the stable sets satisfies di ≥ n − i + 2, 1 ≤ i ...

We prove a number of Turán and Ramsey type stability results for cycles, in particular, the following one: Let n be sufficiently large, 0 < β < 2 × 10 −7 , and the edges of K ⌊(2−β)n⌋ be 2-colored so that no mononchromatic C n exists. Then, for some q ∈ ((1 − β) n, n) , we may drop a vertex v so that in K ⌊(2−β)n⌋ −v one of the colors induces K q,⌊(2−β)n⌋−q−1 , while the other one induces K q ∪...

A sequence S is potentially Km−Pk graphical if it has a realization containing a Km −Pk as a subgraph. Let σ(Km −Pk, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Km−Pk, n) is potentially Km−Pk graphical. In this paper, we prove that σ(Km−Pk, n) ≥ (2m−6)n−(m−3)(m−2)+2, for n ≥ m ≥ k + 1 ≥ 4. We conjecture that equality holds for n ≥ m ≥ k + 1 ≥ 4. W...

Abstract argumentation offers an appealing way of representing and evaluating arguments and counterarguments. This approach can be enhanced by a probability assignment to each argument. There are various interpretations that can be ascribed to this assignment. In this paper, we regard the assignment as denoting the belief that an agent has that an argument is justifiable, i. e., that both the p...