Hoffmann-Ostenhof's Conjecture states that the edge set of every connected cubic graph can be decomposed into a spanning tree, a matching and a 2-regular subgraph. In this paper, we show that the conjecture holds for claw-free subcubic graphs and 4-chordal subcubic graphs. induced subgraph isomorphic to K 1,3 . A cycle is called chordless if it has no chord. A graph G is called chordal if every cycle of G of length greater than 3 has a chord and a graph is 4-chordal if it has no induced cycle of length greater than 4. A cut-edge of a connected graph G is an edge e ∈ E(G) such that G \ e is disconnected. A subdivision of a graph G is a graph obtained from G by replacing some of the edges of G by internally vertex-disjoint paths.Hoffmann-Ostenhof proposed the following conjecture in his thesis [5], this conjecture also was appeared as a problem of BCC22 [4].Hoffmann-Ostenhof 's Conjecture. Let G be a connected cubic graph. Then the edges of G can be decomposed into a spanning tree, a matching and a 2-regular subgraph.An edge decomposition of a graph G is called a good decomposition, if the edges of G can be decomposed into a spanning tree, a matching and a 2-regular subgraph (matching or 2-regular subgraph may be empty). A graph is called good if it has a good decomposition. Hoffmann-Ostenhof's Conjecture is known to be true for some families of cubic graphs. Kostochka [7] showed that the Petersen graph, the prism over cycles, and many other graphs are good.Bachstein [3] proved that every 3-connected cubic graph embedded in torus or Klein-bottle is good. Furthermore, Ozeki and Ye [8] proved that 3-connected cubic plane graphs are good.Akbari et. al. [2] showed that hamiltonian cubic graphs are good. Also, it has been proved that the traceable cubic graphs are good [1]. In 2017, Hoffmann-Ostenhof et. al. [6] proved that planar cubic graphs are good. In this paper it is shown that the connected claw-free subcubic graphs and the connected 4-chordal subcubic graphs are good. Connected Claw-free Subcubic Graphs are GoodIn this section we show that in the case of claw-free graphs, Hoffmann-Ostenhof's Conjecture holds even for claw-free subcubic graphs.Theorem 2.1. If G is a connected claw-free subcubic graph, then G is good.Proof. We apply by induction on n = |V (G)|. If |V (G)| = 3, then the assertion is trivial. Now, we consider the following two cases:Case 1. Assume that the graph G has a cut-edge e. Let H and K be the connected components of G \ e. By induction hypothesis, both H and K are good. Let T i , i = 1, 2, be

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