In this paper, we establish an equivalence between the contractible graphs with respect to the mod (2p + 1)-orientability and the graphs with K 1,2p+1 -decompositions. This is applied to disprove a conjecture proposed by Barat and Thomassen that every 4-edge-connected simple planar graph G with |E(G)| ≡ 0 (mod 3) has a claw decomposition.
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