Path extendable graphs |
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Authors: | G R T Hendry |
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Institution: | (1) Department of Mathematics, Aberdeen University, Edward Wright Building Dunbar Street, Aberdeen, Scotland |
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Abstract: | A pathP in a graphG is said to beextendable if there exists a pathP’ inG with the same endvertices asP such thatV(P)⊆V (P’) and |V(P’)|=|V(P)|+1. A graphG ispath extendable if every nonhamiltonian path inG is extendable. We investigate the extent to which known sufficient conditions for a graph to be hamiltonian-connected imply
the extendability of paths in the graph. Several theorems are proved: for example, it is shown that ifG is a graph of orderp in which the degree sum of each pair of non-adjacent vertices is at leastp+1 andP is a nonextendable path of orderk inG thenk≤(p+1)/2 and 〈V (P)〉≅K
k
orK
k
−e. As corollaries of this we deduce that if δ(G)≥(p+2)/2 or if the degree sum of each pair of nonadjacent vertices inG is at least (3p−3)/2 thenG is path extendable, which strengthen results of Williamson 13]. |
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Keywords: | Primary 05C38 Secondary 05C45 |
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