PARTITIONING OF VERTICES INTO ODD INDUCED SUBGRAPHS

A graph G is called an odd (even) graph if for every vertex v ∈ V (G), d G (v) is odd (even). The subgraph H of G is called an odd (even) induced subgraph if for every vertex v ∈ V (H), d H (v) is odd (even). There is a long history to prove that we can partition any graph G into two even induced subgraphs. Also it was proved that we can partition any graph into two subgraphs which one is even and the other is odd. Our purpose on this paper is to partition a graph of even order into odd induced sub- graphs. We denote the minimum number of odd induced subgraphs which partition G by od(G). It was proved that the vertices of every connected graph of even order can be partitioned into some odd induced subgraphs. In this paper, we show that for every tree T, od(T) ≤ 2 and for every unicyclic graph G, od(G) ≤ 3. Also we show that for every positive integer k, there exists a graph of even order that od(G) k. Moreover, we show that if G is a connected planar graph of even order, then od(G) ≤ 4 and if G is an outer planar graph, od(G) ≤ 3.