We give various reformulations of the Strong Perfect Graph Conjecture, based on a study of forced coloring procedures, uniquely colorable subgraphs and ! ? 1-cliques in minimal imperfect graphs.

A graph is triangulated if it has no chordless cycle with at least four vertices (8k 4; C k 6 6 G). These graphs have been generalized by R. Hayward with the weakly triangulated graphs (8k 5; C k ; C k 6 6 G). In this note we propose a new generalization of triangulated graphs. A graph G is slightly triangulated if it satisses the two following conditions : 1. G contains no chordless cycle with...

We generalize the notion of Jacobson graphs into $n$-array columns called $n$-array Jacobson graphs and determine their connectivities and diameters. Also, we will study forbidden structures of these graphs and determine when an $n$-array Jacobson graph is planar, outer planar, projective, perfect or domination perfect.

Let $G=(V,E)$ be a simple connected graph. A perfect matching (or Kekul'e structure in chemical literature) of $G$ is a set of disjoint edges which covers all vertices of $G$. The anti-forcing number of $G$ is the smallest number of edges such that the remaining graph obtained by deleting these edges has a unique perfect matching and is denoted by $af(G)$. In this paper we consider some specifi...

Certain subgraphs of a given graph G restrict the minimum number χ(G) of colors that can be assigned to the vertices of G such that the endpoints of all edges receive distinct colors. Some of such subgraphs are related to the celebrated Strong Perfect Graph Theorem, as it implies that every graph G contains a clique of size χ(G), or an odd hole or an odd anti-hole as an induced subgraph. In thi...

Normal graphs are defined in terms of cross-intersecting set families: a graph is normal if it admits a clique cover Q and a stable set cover S s.t. every clique in Q intersects every stable set in S. Normal graphs can be considered as closure of perfect graphs by means of co-normal products (K ̈orner [6]) and graph entropy (Czisz ́ar et al. [5]). Perfect graphs have been recently characterized a...

The dichromatic number of D, denoted by χ→(D), is the smallest integer k such that D admits an acyclic k-coloring. We use maderχ→(F) to denote if χ→(D)≥k, then contains a subdivision F. A digraph F called Mader-perfect for every subdigraph F′ F, maderχ→(F′)=|V(F′)|. extend octi digraphs larger class and prove it Mader-perfect, which generalizes result Gishboliner, Steiner Szabó [Dichromatic for...