‎Let G be a graph‎. ‎A 2-rainbow dominating function (or‎ 2-RDF) of G is a function f from V(G)‎ ‎to the set of all subsets of the set {1,2}‎ ‎such that for a vertex v ∈ V (G) with f(v) = ∅, ‎the‎‎condition $bigcup_{uin N_{G}(v)}f(u)={1,2}$ is fulfilled‎, wher NG(v)  is the open neighborhood‎‎of v‎. ‎The weight of 2-RDF f of G is the value‎‎$omega (f):=sum _{vin V(G)}|f(v)|$‎. ‎The 2-rainbow‎‎d...

This column is devoted to maximum (respectively, maximum weight) independent set problems in geometric intersection graphs. We illustrate with one example in each class: (I) The following question was asked by T. Rado in 1928: What is the largest number c such that, for any finite set F of axis-parallel squares in the plane, there exists an independent subset I ⊆ F of pairwise disjoint squares ...

The problem I am going to comment reached me in 1987 at Memphis in a letter of Uncle Paul. He wrote: 'We have the following problem with Hajnal. If G(n) has n points and does not contain induced C4, is it true that it has either a clique or an independent set with n 10 points? Kind regards to your boss+ colleagues, kisses to the E -s.' E.P.' After noting that E have been used in different conte...

Graphs with large minimum degree containing no copy of a clique on r vertices (Kr) must contain relatively large independent sets. A classical result of Andrásfai, Erdős, and Sós implies that Kr-free graphs G with degree larger than ((3r−7)/(3r− 4))|V (G)| must be (r− 1)-partite. An obvious consequence of this result is that the same degree threshold implies an independent set of order (1/(r − ...

Recently, in conversation with Erdős, Hajnal asked whether or not for any triangle-free graph G on the vertex set N, there always exists a sequence 〈xn〉n=1 so that whenever F and H are distinct finite nonempty subsets of N, {Σn∈F xn,Σn∈H xn} is not an edge of G (that is, FS(〈xn〉n=1) is an independent set). We answer this question in the negative. We also show that if one replaces the assumption...

Fifty years ago Erdős asked to determine the minimum number of k-cliques in a graph on n vertices with independence number less than l. He conjectured that this minimum is achieved by the disjoint union of l− 1 complete graphs of size n l−1 . This conjecture was disproved by Nikiforov who showed that the balanced blow-up of a 5-cycle has fewer 4-cliques than the union of 2 complete graphs of si...