Geodesic
Vertex partitioning of graphs into odd induced subgraphs
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 2, pp. 385-399

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A graph G is called an odd (even) graph if for every vertex v∈ V(G), d_G(v) is odd (even). Let G be a graph of even order. Scott in 1992 proved that the vertices of every connected graph of even order can be partitioned into some odd induced forests. We denote the minimum number of odd induced subgraphs which partition V(G) by od(G). If all of the subgraphs are forests, then we denote it by od_F(G). In this paper, we show that if G is a connected subcubic graph of even order or G is a connected planar graph of even order, then od_F(G)≤ 4. Moreover, we show that for every tree T of even order od_F(T)≤ 2 and for every unicyclic graph G of even order od_F(G)≤ 3. Also, we prove that if G is claw-free, then V(G) can be partitioned into at most Δ(G)-1 induced forests and possibly one independent set. Furthermore, we demonstrate that the vertex set of the line graph of a tree can be partitioned into at most two odd induced subgraphs and possibly one independent set.
Keywords: odd induced subgraph, subcubic graphs, claw-free graphs, line graphs, independent set 
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     title = {Vertex partitioning of graphs into odd induced subgraphs},
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Aashtab, Arman; Akbari, Saieed; Ghanbari, Maryam; Shidani, Amitis. Vertex partitioning of graphs into odd induced subgraphs. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 2, pp. 385-399. http://geodesic.mathdoc.fr/item/DMGT_2023_43_2_a4/