# Two Improvements on the Erdös, Harary and Klawe Conjecture

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## Abstract

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 *m* ≥ *ϕ*(*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*) ≤ 2*n* − 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## Keywords

Residual-graph cartesian product composite graph joining of graphs## Preview

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## References

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