We determine the independence number of the Kneser graph on line-plane flags in the projective space PG(4, q).

Let $R$ be a commutative ring. In this paper, by using algebraic properties of $R$, we study the Hase digraph of prime ideals of $R$.

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part I, Discrete Appl. Math., 132, 2004, 97–108] that there are no 5-connected planar w...

A graph is well-covered if every maximal independent set has the same cardinality, namely the vertex independence number. We answer a question of Topp and Volkmann [5] and prove that if the Cartesian product of two graphs is well-covered, then at least one of them must be well-covered.

A d-graph G = (V ; E1, . . . , Ed) is a complete graph whose edges are arbitrarily partitioned into d subsets (colored with d colors); G is a Gallai d-graph if it contains no three-colored triangle ∆; furthermore, G is a CIS d-graph if ⋂d i=1 Si 6= ∅ for every set-family S = {Si | i ∈ [d]}, where Si ⊆ V is a maximal independent set of Gi = (V,Ei), the ith chromatic component of G, for all i ∈ [...

A general partition graph (gpg) is an intersection graph G on a set S so that for every maximal independent set M of vertices in G, the subsets assigned to the vertices in M partition S. These graphs have been characterized by the presence of special clique covers. The Triangle Condition T for a graph G is that for any maximal independent set M and any edge uv in G M, there is a vcrtex W E M so...

We study different classes of digraphs, which are generalizations of tournaments, to have the property of possessing a maximal independent set intersecting every non-augmentable path (in particular, every longest path). The classes are the arclocal tournament, quasi-transitive, locally in-semicomplete (out-semicomplete), and semicomplete k-partite digraphs. We present results on strongly intern...

The square G2 of a graph G is defined on the vertex set of G in such a way that distinct vertices with distance at most two in G are joined by an edge. We study the chromatic number of the square of the Cartesian product Cm2Cn of two cycles and show that the value of this parameter is at most 7 except when m = n = 3, in which case the value is 9, and when m = n = 4 or m = 3 and n = 5, in which ...

A theorem of O’Nan and Scott [6]; [2, Chapter 4] restricts the possibilities for maximal subgroups of finite symmetric groups: they are of six types of which the first four are explicitly known, the fifth involves a finite simple group, and the sixth an action of a simple group. This result, in conjunction with the Classification of Finite Simple Groups, has a number of consequences. In particu...