D λ -cycles in 3-cyclable graphs

Abstract

Letk andλ be positive integers, andG a 2-connected graph of ordern with minimum degreeδ and independence numberα. A cycleC ofG is called aD λ-cycle if every component ofG − V(C) has order smaller thanλ. The graphG isk-cyclable if anyk vertices ofG lie on a common cycle. A previous result of the author is that ifλ ≥ k ≥ 2, G isk-connected and every connected subgraphH ofG of orderλ has at leastn +k 2 + 1/k + 1 −λ vertices outsideH adjacent to at least one vertex ofH, thenG contains aD λ-cycle. Here it is conjectured that “k-connected” can be replaced by “k-cyclable”, and this is proved fork = 3. As a consequence it is shown that ifn ≤ 4δ − 6, or ifG is triangle-free andn ≤ 8δ − 10, thenG contains aD 3-cycle orG ∈ ℱ, whereℱ denotes a well-known class of nonhamiltonian graphs of connectivity 2. As an analogue of a result of Nash-Williams it follows that ifn ≤ 4δ − 6 andα ≤δ − 1, thenG is hamiltonian orG ∈ ℱ. The results are all best possible and compare favorably with recent results on hamiltonicity of graphs which are “close to claw-free”.