Vertex partitioning of graphs into odd induced subgraphs

Arman Aashtab, Saieed Akbari, Maryam Ghanbari, Amitis Shidani

Abstract: A graph $G$ is called an odd (even) graph if for every vertex $v\in V(G)$, $d_G(v)$ 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 $od_F(G)\le 4$. Moreover, we show that for every tree $T$ of even order $od_F(T)\le 2$ and for every unicyclic graph $G$ of even order $od_F(G)\le 3$. Also, we prove that if $G$ is claw-free, then $V(G)$ can be partitioned into at most $\Delta(G)-1$ 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.