For an edge-colored graph G , we call edge–cut M of monochromatic if the edges are colored with same color. The is called monochromatically disconnected any two distinct vertices separated by a edge–cut. connected disconnection number denoted m d ( ) maximum colors that needed in order to make disconnected. We show almost all graphs have numbers equal 1. also obtain Nordhaus–Gaddum-type results...

We extend the Grundy number and ochromatic number, parameters on graph colorings, to digraph we call them digrundy diochromatic , respectively. First, prove that for every equals (as happens graphs). Then, interpolation property Nordhaus–Gaddum relations improve dichromatic diachromatic numbers bounded previously by authors in Araujo-Pardo et al. (2018).

For each vertex s of the vertex subset S of a simple graph G, we define Boolean variables p = p(s, S), q = q(s, S) and r = r(s, S) which measure existence of three kinds of S-private neighbours (S-pns) of s. A 3-variable Boolean function f = f(p, q, r) may be considered as a compound existence property of S-pns. The subset S is called an f -set of G if f = 1 for all s ∈ S and the class of f -se...

A k-decomposition (G1, . . . , Gk) of a graph G is a partition of its edge set to form k spanning subgraphs G1, . . . , Gk. The classical theorem of Nordhaus and Gaddum bounds χ(G1) + χ(G2) and χ(G1)χ(G2) over all 2-decompositions of Kn. For a graph parameter p, let p(k;G) denote the maximum of ∑k i=1 p(Gi) over all k-decompositions of the graph G. The clique number ω, chromatic number χ, list ...

A set S of vertices of a graph G is a connected dominating set if every vertex not in S is adjacent to some vertex in S and the subgraph induced by S is connected. The connected domination number γc(G) is the minimum size of a connected dominating set of G. In this paper we prove that γc(G) + γc(G) ≤ min{δ(G), δ(G)} + 4 for every n-vertex graph G such that G and G have diameter 2 and show that ...

The chromatic vertex (resp. edge) stability number vsχ(G) esχ(G)) of a graph G is the minimum vertices edges) whose deletion results in H with χ(H)=χ(G)−1. In main result it proved that if χ(G)∈{Δ(G),Δ(G)+1}, then vsχ(G)=ivsχ(G), where ivsχ(G) independent number. need not hold for graphs χ(G)≤Δ(G)+12. It χ(G)>Δ(G)2+1, vsχ(G)=esχ(G). A Nordhaus–Gaddum-type on also given.

Let !:ten) denote the class of simple graphs of order n. In this paper we consider a variation of this problem by restricting our attention to the subclass of !:ten) consisting of graphs having exactly m edges. We consider the parameters edge connectivity, diameter and chromatic number. We also consider the problem of characterizing the extremal graphs and the realizability problem. 1. INTRODUC...