LetF be a field of characteristic p 6= 0 andG an Abelian group. For each prime q and each ordinal α we calculate in terms of G and its sections the Warfield q-invariantsWα,q(V (FG)) of the group V (FG) of all normalized units in the commutative group algebra FG when either q = p,Gt/Gp is infinite and F is perfect, orG is p-mixed. Surprisingly, these invariants do not depend on F . This supplies...

Roussel and Rubio proved a lemma which is essential in the proof of the Strong Perfect Graph Theorem. We give a new short proof of the main case of this lemma. In this note, we also give a short proof of Hayward’s decomposition theorem for weakly chordal graphs, relying on a Roussel–Rubio-type lemma. We recall how Roussel–Rubio-type lemmas yield very short proofs of the existence of even pairs ...

LetD = (V (D), A(D)) be a digraph; a kernelN ofD is a set of verticesN ⊆ V (D) such that N is independent (for any x, y∈N there is no arc between them) and N is absorbent (for each x ∈ V (D) −N) there exists an xN -arc in D). A digraph D is said to be kernel-perfect whenever each one of its induced subdigraphs has a kernel. A digraph D is oriented by sinks when every semicomplete subdigraph of ...

A linear (qd, q, t)-perfect hash family of size s in a vector space V of order qd over a field F of order q consists of a set φ1, . . . , φs of linear functionals from V to F with the following property: for all t subsets X ⊆ V there exists i ∈ {1, . . . , s} such that φi is injective when restricted to F . A linear (qd, q, t)-perfect hash family of minimal size d(t− 1) is said to be optimal. I...

We will extend Reed's Semi-Strong Perfect Graph Theorem by proving that unbreakable C 5-free graphs diierent from a C 6 and its complement have unique P 4-structure.

The study of nearly perfect sets in graphs was initiated in [2]. Let S ⊆ V (G). We say that S is a nearly perfect set (or is nearly perfect) in G if every vertex in V (G)−S is adjacent to at most one vertex in S. A nearly perfect set S in G is called maximal if for every vertex u ∈ V (G) − S, S ∪ {u} is not nearly perfect in G. The minimum cardinality of a maximal nearly perfect set is denoted ...

Let k, v, t be integers such that k ≥ v ≥ t ≥ 2. A perfect hash family PHF(N ; k, v, t) can be defined as an N × k array with entries from a set of v symbols such that every N× t subarray contains at least one row having distinct symbols. Perfect hash families have been studied by over 20 years and they find a wide range of applications in computer sciences and in cryptography. In this paper we...

A subset S ⊆ V in a graph G = (V,E) is a [1, 2]-set if for every vertex v ∈ V \ S, 1 ≤ |N(v)∩ S| ≤ 2, that is, every vertex v ∈ V \ S is adjacent to at least one but not more than two vertices in S. In this paper we relate the concept of [1, 2]-sets to a host of other concepts in domination theory, including perfect domination, efficient domination, nearly perfect sets, 2-packings and k-depende...

Given a graphG = (V,E) and a set Lv of admissible colors for each vertex v ∈ V (termed the list at v), a list coloring of G is a (proper) vertex coloring φ : V → ⋃ v∈V Lv such that φ(v) ∈ Lv for all v ∈ V and φ(u) 6= φ(v) for all uv ∈ E. If such a φ exists, G is said to be list colorable. The choice number of G is the smallest natural number k for which G is list colorable whenever each list co...

Let k be a positive integer. A vertex subset D of a graph G = (V,E) is a perfect k-dominating set of G if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γkp(G). In this paper, we generalize perfect domination to perfect k-domination, where many bounds of γkp(G) are obtained. We ...