We consider degree sum conditions and the existence of vertex-disjoint cycles in a graph. In this paper, we prove the following: Suppose that G is a graph of order at least 3k + 2 and σ3(G) ≥ 6k − 2, where k ≥ 2. Then G contains k vertex-disjoint cycles. The degree and order conditions are sharp.

For given positive integers j ≥ k, an L( j, k)-labeling of a graph G is a function f : V (G) → {0, 1, 2, . . .} such that | f (u) − f (v)| ≥ j when dG (u, v) = 1 and | f (u) − f (v)| ≥ k when dG (u, v) = 2. The L( j, k)-labeling number λ j,k(G) of G is defined as the minimum m such that there is an L( j, k)-labeling f of G with f (V (G)) ⊆ {0, 1, 2, . . . ,m}. For a graph G of maximum degree ∆ ...

Let wk be the minimum degree sum of a path on k vertices in a graph. We prove for normal plane maps that: (1) if w2 = 6, then w3 may be arbitrarily big, (2) if w2 > 6, then either w3 ≤ 18 or there is a ≤ 15-vertex adjacent to two 3-vertices, and (3) if w2 > 7, then w3 ≤ 17.

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 it 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 + 1, ...

In this paper characterizations of connected unicyclic and bicyclic graphs in terms of the degree sequence, as well as the graphs in these classes minimal with respect to the degree distance are given. © 2007 Elsevier B.V. All rights reserved.

We show that the number of distinct non-parallel lines passing through two conjugates of an algebraic number α of degree d ≥ 3 is at most [d/2] − d + 2, its conjugates being in general position if this number is attained. If, for instance, d ≥ 4 is even, then the conjugates of α ∈ Q of degree d are in general position if and only if α has 2 real conjugates, d− 2 complex conjugates, no three dis...

Smooth complex surfaces polarized with an ample and globally generated line bundle of degree three and four, such that the adjoint bundle is not globally generated, are considered. Scrolls of a vector bundle over a smooth curve are shown to be the only examples in degree three. Two classes of examples in degree four are presented, one of which is shown to characterize regular such pairs. A Reid...

The boxicity of a graph G = (V,E) is the smallest k for which there exist k interval graphs Gi = (V,Ei), 1 ≤ i ≤ k, such that E = E1 ∩ . . . ∩ Ek. Graphs with boxicity at most d are exactly the intersection graphs of (axis-parallel) boxes in Rd. In this note, we prove that graphs with maximum degree ∆ have boxicity at most ∆2 + 2, which improves the previous bound of 2∆2 obtained by Chandran et...

A conjecture of Grone and Merris states that for any graph G, its Laplacian spectrum, Λ(G), is majorized by its conjugate degree sequence, D∗(G). That conjecture prompts an investigation of the relationship between Λ(G) and D∗(G), and Merris has characterized the graphs G for which the multisets Λ(G) and D∗(G) are equal. In this paper, we provide a constructive characterization of the graphs G ...

For any enumeration degree a let D a be the set of s-degrees contained in a. We answer an open question of Watson by showing that if a is a nontrivial Σ2-enumeration degree, then D a has no least element. We also show that every countable partial order embeds into D a.