Path factors in cubic graphs

Abstract

Abstract Let ℱ be a set of connected graphs. An ℱ‐factor of a graph is its spanning subgraph such that each component is isomorphic to one of the members in ℱ. Let P k denote the path of order k . Akiyama and Kano have conjectured that every 3‐connected cubic graph of order divisible by 3 has a { P 3 }‐factor. Recently, Kaneko gave a necessary and sufficient condition for a graph to have a { P 3 , P 4 , P 5 }‐factor. As a corollary, he proved that every cubic graph has a { P 3 , P 4 , P 5 }‐factor. In this paper, we prove that every 2‐connected cubic graph of order at least six has a { P k ∣ k ≥ , 6}‐factor, and hence has a { P 3 , P 4 }‐factor. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 188–193, 2002; DOI 10.1002/jgt.10022