The Classification Problem is the problem of deciding whether a simple graph has chromatic index equals to ∆ or ∆+1, where ∆ is the maximum degree of the graph. It is known that to decide if a graph has chromatic index equals to ∆ is NP-complete. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. The chromatic indexes for some subsets of split graphs, s...

A graph G = (V,E) is a unipolar graph if there exits a partition V = V1 ∪ V2 such that, V1 is a clique and V2 induces the disjoint union of cliques. The complement-closed class of generalized split graphs are those graphs G such that either G or the complement of G is unipolar. Generalized split graphs are a large subclass of perfect graphs. In fact, it has been shown that almost all C5-free (a...

A λ-coloring of a graph G is an assignment of colors from the integer set {0, . . . , λ} to the vertices of the graph G such that vertices at distance at most two get different colors and adjacent vertices get colors which are at least two apart. The problem of finding λ-coloring with small or optimal λ arises in the context of radio frequency assignment. We show that the problem of finding the...

Spider graphs are the intersection graphs of subtrees of subdivisions of stars. Thus, spider graphs are chordal graphs that form a common superclass of interval and split graphs. Motivated by previous results on the existence of Hamilton cycles in interval, split and chordal graphs, we show that every 3/2-tough spider graph is hamiltonian. The obtained bound is best possible since there are (3/...

Kőnig-Egerváry graphs (KEGs) are the graphs whose maximum size of a matching is equal to the minimum size of a vertex cover. We give an excluded subgraph characterization of KEGs. We show that KEGs are a special case of a more general class of graph: Red/Blue-split graphs, and give an excluded subgraph characterization of Red/Blue-split graphs. We show several consequences of this result includ...

A λ-coloring of a graph G is an assignment of colors from the integer set {0, . . . , λ} to the vertices of the graph G such that vertices at distance of at most two get different colors and adjacent vertices get colors which are at least two apart. The problem of finding λ-colorings with optimal or near-optimal λ arises in the context of radio frequency assignment. We show that the problem of ...

The families EPT (resp. EPG) Edge Intersection Graphs of Paths in a tree (resp. in a grid) are well studied graph classes. Recently we introduced the graph classes Edge-Intersecting and Non-Splitting Paths in a Tree (ENPT), and in a Grid (ENPG). It was shown that ENPG contains an infinite hierarchy of subclasses that are obtained by restricting the number of bends in the paths. Motivated by thi...

We consider several graphs classes defined in terms of conditions on cliques and stable sets, including CIS, split, equistable, and other related classes. We pursue a systematic study of the relations between them. As part of this study, we introduce two generalizations of CIS graphs, obtain a new characterization of split graphs, and a characterization of CIS line graphs.

Finding the pebbling number of a graph is harder than NP-complete (Π 2 complete, to be precise). However, for many families of graphs there are formulas or polynomial algorithms for computing pebbling numbers; for example, complete graphs, products of paths (including cubes), trees, cycles, diameter two graphs, and more. Moreover, graphs having minimum pebbling number are called Class 0, and ma...

An edge-coloring of a graph is an assignment of colors to its edges such that no adjacent edges have the same color. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. Split graphs have been introduced by Földes and Hammer [4] and it is a well-studied class of graphs. However, the problem of deciding the chromatic index of any split graph remains unsolv...