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We consider cubic graphs formed with k ≥ 2 disjoint claws $C_i{K}_{1,3}$ (0 ≤ i ≤ k-1) such that for every integer i modulo k the three vertices of degree 1 of $C_i$ are joined to the three vertices of degree 1 of $C_{i-1}$ and joined to the three vertices of degree 1 of $C_{i+1}$. Denote by $t_i$ the vertex of degree 3 of $C_i$ and by T the set $t₁,t₂,...,t_{k-1}$. In such a way we construct three distinct graphs, namely FS(1,k), FS(2,k) and FS(3,k). The graph FS(j,k) (j ∈ 1,2,3) is the graph where the set of vertices ${\bigcup }_{i=0}^{i=k-1}V\left(C_i\right)\setminus T$ induce j cycles (note that the graphs...

A linear forest is a graph whose connected components are chordless paths. A linear partition of a graph G is a partition of its edge set into linear forests and la(G) is the minimum number of linear forests in a linear partition. In this paper we consider linear partitions of cubic simple graphs for which it is well known that la(G) = 2. A linear partition $L=(L_B,L_R)$ is said to be odd whenever each path of $L_B\cup L_R$ has odd length and semi-odd whenever each path of $L_B$ (or each path of $L_R$) has odd length. In [2] Aldred...