let f, g and h be non-empty graphs. the notation f → (g,h) means that if any edge of f is colored by red or blue, then either the red subgraph of f con- tains a graph g or the blue subgraph of f contains a graph h. a graph f (without isolated vertices) is called a ramsey (g,h)−minimal if f → (g,h) and for every e ∈ e(f), (f − e) 9 (g,h). the set of all ramsey (g,h)−minimal graphs is denoted by ...

Let F be a family of graphs. In the F-Completion problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k edges can be added to G so that the resulting graph does not contain a graph from F as an induced subgraph. It appeared recently that special cases of F-Completion, the problem of completing into a chordal graph known as Minimum Fill-in, corresponding...

We provide a formula for the number of edges of a maximum induced matching in a graph. As applications, we give some structural properties of (k + 1 )K2-free graphs, construct all 2K2-free graphs, and count the number of labeled 2K2-free connected bipartite graphs.

A graph is called 2K2-free if it does not contain two independent edges as an induced subgraph. Mou and Pasechnik conjectured that every 3 2 -tough 2K2-free graph with at least three vertices has a spanning trail with maximum degree at most 4. In this paper, we confirm this conjecture. We also provide examples for all t < 54 of t-tough graphs that do not have a spanning trail with maximum degre...

A graph is 2K2-partitionable if its vertex set can be partitioned into four nonempty parts A, B, C , D such that each vertex of A is adjacent to each vertex of B, and each vertex of C is adjacent to each vertex of D. Determining whether an arbitrary graph is 2K2-partitionable is the only vertex-set partition problem into four nonempty parts according to external constraints whose computational ...

We call a graph 2K2-free if it is connected and does not contain two independent edges as an induced subgraph. The assumption of connectedness in this definition only serves to eliminate isolated vertices. Wagon [6] proved that x(G) ~ w(G)[w(G) + 1]/2 if G is 2Krfree where x(G) and w(G) denote respectively the chromatic number and maximum clique size of G. Further properties of 2K2-free graphs ...

A hereditary class G of graphs is χ-bounded if there is a χ-binding function, say f such that χ(G) ≤ f(ω(G)), for every G ∈ G, where χ(G) (ω(G)) denote the chromatic (clique) number of G. It is known that for every 2K2-free graph G, χ(G) ≤ ( ω(G)+1 2 ) , and the class of (2K2, 3K1)-free graphs does not admit a linear χ-binding function. In this paper, we are interested in classes of 2K2-free gr...

A complete graph is the graph in which every two vertices are adjacent. For a graph G = (V,E), the complete width of G is the minimum k such that there exist k independent sets Ni ⊆ V , 1 ≤ i ≤ k, such that the graph G obtained from G by adding some new edges between certain vertices inside the sets Ni, 1 ≤ i ≤ k, is a complete graph. The complete width problem is to decide whether the complete...

We give an O(n) algorithm to find a minimum clique cover of a (bull, C4)-free graph, or equivalently, a minimum colouring of a (bull, 2K2)-free graph, where n is the number of vertices of the graphs.

Let F, G and H be non-empty graphs. The notation F → (G,H) means that if any edge of F is colored by red or blue, then either the red subgraph of F con- tains a graph G or the blue subgraph of F contains a graph H. A graph F (without isolated vertices) is called a Ramsey (G,H)−minimal if F → (G,H) and for every e ∈ E(F), (F − e) 9 (G,H). The set of all Ramsey (G,H)−minimal graphs is denoted by ...