The proper vertex-disconnection of graphs

Abstract

Let G be a vertex-colored connected graph. A subset X of the vertex-set of G is called proper if any two adjacent vertices in X have distinct colors. The graph G is called proper vertex-disconnected if for any two vertices x and y of G, there exists a vertex subset S of G such that when x and y are nonadjacent, S is proper and x and y belong to different components of G−S; whereas when x and y are adjacent, S+x or S+y is proper and x and y belong to different components of (G−xy)−S. For a connected graph G, the proper vertex-disconnection number of G, denoted by pvd(G), is the minimum number of colors that are needed to make G proper vertex-disconnected.In this paper, we firstly characterize the graphs of order n with proper vertex-disconnection number k for k∈{1,n−2,n−1,n}. Secondly, we give some sufficient conditions for a graph G such that pvd(G)=χ(G), and show that almost all graphs G have pvd(G)=χ(G) and pvd(G‾)=χ(G‾). We also give an equivalent statement of the famous Four Color Theorem. Furthermore, we study the relationship between the proper disconnection number pd(G) of G and the proper vertex-disconnection number pvd(L(G)) of the line graph L(G) of G. Finally, we show that it is NP-complete to decide whether a given vertex-colored graph is proper vertex-disconnected, and it is NP-hard to decide for a fixed integer k≥3, whether the pvd-number of a graph G is no more than k, even if k=3 and G is a planar graph with Δ(G)=12. We also show that it is solvable in polynomial time to determine the proper vertex-disconnection number for a graph with maximum degree less than four.