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A sufficient condition for graphs with large neighborhood unions to be traceable. (English) Zbl 0869.05041
Author’s abstract: We prove that a 2-connected graph $$G$$ of order $$p$$ is traceable if $$|N(u)\cup N(v)|+|N(w)\cup N(x)|\geq p-1$$ for all 4-tupes $$\{u,v,w,x\}$$ with $$d(u,v)= d(w,x)= 2$$ $$(u,v,w,x$$ are distinct vertices of $$G)$$. In addition, we give a short proof of Lindquester’s conjecture, see T. E. Lindquester [J. Graph Theory 13, No. 3, 335-352 (1989; Zbl 0695.05042)].
Author’s note added in proof: After the submission of this paper to Discrete Mathematics, I became aware of the manuscript of Professors J. Li and F. Tian, ‘A proof of a conjecture about $$D_\lambda$$-paths in graphs with large neighborhood unions’ on a very similar subject to mine. The paper appears in this volume of Discrete Mathematics on pages 185-196 [see Zbl 0869.05040 above].

##### MSC:
 05C45 Eulerian and Hamiltonian graphs 05C38 Paths and cycles 05C12 Distance in graphs
##### Citations:
Zbl 0695.05042; Zbl 0869.05040
Full Text:
##### References:
 [1] Chartrand, G.; Lesniak, L., Graphs & digraphs, (1986), Wadsworth & Brooks/Cole Monterrey, CA · Zbl 0666.05001 [2] Lindquester, T.E., The effect of distance and neighborhood union conditions on Hamiltonian properties in graphs, J. graph theory, 13, 335-352, (1989) · Zbl 0695.05042
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