A sufficient condition for graphs with large neighborhood unions to be traceable.

*(English)*Zbl 0869.05041Author’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].

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].

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##### 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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