For k ≥ 3, a k-angulation is a 2-connected plane graph in which every internal face is a k-gon. We say that a point set P admits a plane graph G if there is a straight-line drawing of G that maps V (G) onto P and has the same facial cycles and outer face as G. We investigate the conditions under which a point set P admits a k-angulation and find that, for sets containing at least 2k2 points, th...

A chain probe graph is a graph that admits an independent set S of vertices and a set F of pairs of elements of S such that G+ F is a chain graph (i.e., a 2K2-free bipartite graph). We show that chain probe graphs are exactly the bipartite graphs that do not contain as an induced subgraph a member of a family of six forbidden subgraphs, and deduce an O(n2) recognition algorithm. Given a class C...

In an edge deletion problem one is asked to delete at most k edges from a given graph such that the resulting graph satisfies a certain property. In this work, we study four NP-complete edge deletion problems where the goal graph has to be a chain, a split, a threshold, or a co-trivially perfect graph, respectively. All these four graph classes are characterized by a common forbidden induced su...

The reconfiguration graph for the k-colourings of a G, denoted Rk(G), is whose vertices are G and two colourings joined by an edge if they differ in colour on exactly one vertex. For any k-colourable P4-free Bonamy Bousquet proved that Rk+1(G) connected. In this short note, we complete classification connectedness excluding fixed path, constructing 7-chromatic 2K2-free (and hence P5-free) admit...

Using computer algorithms we establish that the Ramsey number R(3,K10− e) is equal to 37, which solves the smallest open case for Ramsey numbers of this type. We also obtain new upper bounds for the cases of R(3,Kk − e) for 11 ≤ k ≤ 16, and show by construction a new lower bound 55 ≤ R(3,K13 − e). The new upper bounds on R(3,Kk − e) are obtained by using the values and lower bounds on e(3,Kl − ...

In 1966, Gallai asked whether all longest paths in a connected graph have nonempty intersection. The answer to this question is not true general and various counterexamples been found. However, there positive solution Gallai’s for many well-known classes of graphs such as split graphs, series–parallel 2K2-free graphs. Split were proven be Hamiltonian under given toughness conditions. This obser...

The class of 2K2-free graphs and its various subclasses have been studied in a variety of contexts. In this paper, we are concerned with the colouring of (P3 ∪P2)-free graphs, a super class of 2K2-free graphs. We derive a O(ω3) upper bound for the chromatic number of (P3 ∪P2)-free graphs, and sharper bounds for (P3 ∪P2, diamond)-free graphs, where ω denotes the clique number. By applying simila...

The contact graph of an arbitrary finite packing of unit balls in Euclidean 3-space is the (simple) graph whose vertices correspond to the packing elements and whose two vertices are connected by an edge if the corresponding two packing elements touch each other. One of the most basic questions on contact graphs is to find the maximum number of edges that a contact graph of a packing of n unit ...

Let G be a graph. We say that is perfectly divisible if for each induced subgraph H of G, V(H) can partitioned into A and B such H[A] perfect ω(H[B])<ω(H). use Pt Ct to denote path cycle on t vertices, respectively. For two disjoint graphs F1 F2, we F1∪F2 the graph with vertex set V(F1)∪V(F2) edge E(F1)∪E(F2), F1+F2 E(F1)∪E(F2)∪{xy|x∈V(F1) y∈V(F2)}. In this paper, prove (i) (P5,C5,K2,3)-free ar...

A discretized rotation is the composition of an Euclidean rotation with the rounding operation. For 0 < α < π/4, we prove that the discretized rotation [rα] is bijective if and only if there exists a positive integer k such as {cosα, sinα} = { 2k + 1 2k2 + 2k + 1 , 2k + 2k 2k2 + 2k + 1 } The proof uses a particular subgroup of the torus (R/Z).