The threshold dimension of a graph G(V,E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. A k-dimensional box is the Cartesian product R1 ×R2 × · · · × Rk where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection ...

A subgraph of an edge-coloured graph is rainbow if all of its edges have different colours. For graphs G and H the anti-Ramsey number ar(G,H) is the maximum number of colours in an edge-colouring of G with no rainbow copy of H. The notion was introduced by Erdős, Simonovits and V. Sós and studied in case G = Kn. Afterwards exact values or bounds for anti-Ramsey numbers ar(Kn, H) were establishe...

An (r, n)-split coloring of a complete graph is an edge coloring with r colors under which the vertex set is partitionable into r parts so that for each i, part i does not contain K, in color i. This generalizes the notion of split graphs which correspond to (2, 2)-split colorings. The smallest N for which the complete graph Ks has a coloring which is not (r,n)-split is denoted by f,.(n). Balan...

The simple max-cut problem is as follows: given a graph, find a partition of its vertex set into two disjoint sets, such that the number of edges having one endpoint in each set is as large as possible. A split graph is a graph whose vertex set admits a partition into a stable set and a clique. The simple max-cut decision problem is known to be NP-complete for split graphs. An indifference grap...

It is well-known that the GRAPH 3.COLORABILITY problem, deciding whether a given graph has a stable set whose deletion results in a bipartite graph, is NP-complete. We prove the following related theorems: It is NP-complete to decide whether a graph has a stable set whose deletion results in (1) a tree or (2) a trivially perfect graph, and there is a polynomial algorithm to decide if a given gr...

A graph is König-Egerváry if the size of a minimum vertex cover equals the size of a maximum matching in the graph. We show that the problem of deleting at most k vertices to make a given graph König-Egerváry is fixedparameter tractable with respect to k. This is proved using interesting structural theorems on matchings and vertex covers which could be useful in other contexts. We also show an ...

The k-th power H of a graph H is obtained from H by adding new edges between every two distinct vertices having distance at most k in H . Lau [Bipartite roots of graphs, ACM Transactions on Algorithms 2 (2006) 178–208] conjectured that recognizing k-th powers of some graph is NP-complete for all fixed k ≥ 2 and recognizing k-th powers of a bipartite graph is NP-complete for all fixed k ≥ 3. We ...

A graph G = (V, E) is called a split graph if there exists a partition V = I∪K such that the subgraphs G[I] andG[K] of G induced by I andK are empty and complete graphs, respectively. In 1980, Burkard and Hammer gave a necessary condition for a split graph G with |I| < |K| to be hamiltonian. We will call a split graph G with |I| < |K| satisfying this condition a Burkard–Hammer graph. Further, a...