Mediterranean Journal of Mathematics

, Volume 12, Issue 2, pp 263–279 | Cite as

Two Improvements on the Erdös, Harary and Klawe Conjecture



A graph G is said to be F-residual if for every vertex u in G, the graph obtained by removing the closed neighborhood of u from G is isomorphic to F, here F is a given graph. Inductively, define a multiple-F-residual graph by saying that G is m-F-residual if the removal of the closed neighborhood of any vertex of G results in an (m − 1)-F-residual graph. Erdös et al. in Ann Discrete Math 6:117–123, (1980) determined the minimum order of the m-K n -residual graphs for all m and n, which are not necessarily connected, the minimum order of connected; K n -residual graphs, all K n -residual extremal graphs and some canonical connected m-K n -residual graphs. They also stated two conjectures regarding the connected m-K n -residual graphs. It is discussed for the case that m = 2 and n ≥ 4 in Shihui Yang Oper Res 2:71–82, (1984) and Liao et al. in Mediterr J Math 625–641, (2013). In this paper, [Improvement 1] we show that G* = G[K t ] is an m-K nt -residual graph, whenever G is an m-K n -residual graph, and for odd n, C 5[K n ] is the unique K 2n -residual graph with the least odd in Yang and Duan in ACTA Math Appl Sinica 34:765–778, (2011); there is a unique connected 2-K 6-residual graph, which is not isomorphic to K 8 × K 3 of the minimum order. [Improvement 2] we prove that for every n ≥ 3, there is an integer denoted by ϕ(n), for every mϕ(n), the minimum order denoted by ϕ n(m) of connected m-K n -residual graphs is (m+3)n+m−1, and \({G \cong G_{n,1}^{m}}\) is the unique up to isomorphism extremal graph for every m > ϕ(n). Furthermore, we determined that ϕ(3) = 4, ϕ(4) = 7, and ϕ(n) ≤ 2n − 3, n ≥ 5. In particular, we determine the minimum order of connected m-K n -residual graphs for n = 3, 4 and every m ≥ 2, and all the corresponding extremal graphs.

Mathematics Subject Classification (2010)

05C35 05C75 05C60 


Residual-graph cartesian product composite graph joining of graphs 


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Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.College of Mathematics and Computer SciencesYangtze Normal UniversityChongqingPeople’s Republic of China
  2. 2.College of ScienceChongqing Technology and Business UniversityChongqingPeople’s Republic of China

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