Abstract
In 1988 Faudree et al. proved that let G be a 3-connected K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,3</sub> -free graph of order n, if |N(x)∪ N(y)|≥(2n-4)/3 for each pair of nonadjacent vertices x,y, then G is Homogeneously traceable. In 1991nian Bauer et al. proved that let G be a 3-connected K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,3</sub> -free graph of order n, if |N(x) ∪ N(y)|≥(2n-5)/3 for each pair of nonadjacent vertices x,y, then G is traceable. In this note we prove the further result: let G be a 3-connected K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1,3</sub> -free graph of order n, if |N(x) ∪ N(y)|≥(2n-6)/3 for each pair of nonadjacent vertices x,y with 1≤|N(x) ∩ N(y)|≤α-1, then G is Homogeneously traceable.
Published Version
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