1 research outputs found
Pairs of Fan-type heavy subgraphs for pancyclicity of 2-connected graphs
A graph on vertices is Hamiltonian if it contains a spanning cycle,
and pancyclic if it contains cycles of all lengths from 3 to . In 1984, Fan
presented a degree condition involving every pair of vertices at distance two
for a 2-connected graph to be Hamiltonian. Motivated by Fan's result, we say
that an induced subgraph of is -heavy if for every pair of
vertices , implies .
For a given graph , is called --heavy if every induced subgraph
of isomorphic to is -heavy. In this paper we show that for a
connected graph with and a 2-connected claw--heavy graph
which is not a cycle, being --heavy implies is pancyclic if
or , where claw is and is the path
plus the edge . Our result partially
improves a previous theorem due to Bedrossian on pancyclicity of 2-connected
graphs.Comment: 11 pages; 2 figures; accepted by Australasian J. Combi