Group Connectivity of Complementary Graphs

  • Authors:

  • Xinmin Hou;Hong-Jian Lai;Ping Li;Cun-Quan Zhang

  • Affiliations:

  • Department of Mathematics, University of Science and Technology of China, Hefei, 230026, People's Republic of China;College of Mathematics and System Sciences, Xinjiang University, Urumqi, Xinjiang, 830046, People's Republic of China and Department of Mathematics West Virginia University, Morgantown, West Virgi ...;Department of Mathematics West Virginia University, Morgantown, West Virginia 26506;Department of Mathematics West Virginia University, Morgantown, West Virginia 26506

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2012

Quantified Score

Hi-index 0.00

Visualization

Abstract

Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and A* = A−{0}. A graph G is A-connected if G has an orientation D(G) such that for every function b: V(G)↦A satisfying , there is a function f: E(G)↦A* such that for each vertex v∈V(G), the total amount of f values on the edges directed out from v minus the total amount of f values on the edges directed into v equals b(v). For a 2-edge-connected graph G, define Λg(G) = min{k: for any abelian group A with |A|⩾k, G is A-connected }. In this article, we prove the following Ramsey type results on group connectivity: Let G be a simple graph on n⩾6 vertices. If min{δ(G), δ(Gc)}⩾2, then either Λg(G)⩽4, or Λg(Gc)⩽4. Let Z3 denote the cyclic group of order 3, and G be a simple graph on n⩾44 vertices. If min{δ(G), δ(Gc)}⩾4, then either G is Z3-connected, or Gc is Z3-connected. © 2011 Wiley Periodicals, Inc. J Graph Theory © 2012 Wiley Periodicals, Inc.