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- W2795104049 abstract "Abstract Let G = ( V , E ) be a graph of order n, and λ = ( λ 1 , λ 2 , … , λ p ) a sequence of positive integers. The sequence λ is admissible for G if λ 1 + ⋯ + λ p = n . Such an admissible sequence λ is said to be realizable in G if there exists a partition ( V 1 , V 2 , … , V p ) of the vertex set V such that Vi induces a connected subgraph of order ni in G for each i. If every admissible sequence is realizable in G, then we say that G is arbitrarily partitionable (AP, for short). We show that if a tree T of maximum degree at most n + 1 has a path containing all the vertices of degree n + 1 , then T □ C n has a Hamiltonian path. In particular, for any caterpillar T with maximum degree at most n + 1 , T □ C n is AP. In addition, if T is a caterpillar with Δ ( T ) ≥ n + 4 , then T □ C n is not AP. For the cases n + 2 ≤ Δ ( T ) ≤ n + 3 , we present some sufficient conditions for a caterpillar T such that T □ C n is AP." @default.
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- W2795104049 date "2018-09-01" @default.
- W2795104049 modified "2023-10-17" @default.
- W2795104049 title "Partitioning the Cartesian product of a tree and a cycle" @default.
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- W2795104049 doi "https://doi.org/10.1016/j.amc.2018.03.015" @default.
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