Tree-colorable maximal planar graphs 1

A tree-coloring of a maximal planar graph is a proper vertex 4-coloring such that every bichromatic subgraph, induced by this coloring, is a tree. A maximal planar graph G is tree-colorable if G has a tree-coloring. In this article, we prove that a tree-colorable maximal planar graphG with (G) 4 contains at least four odd-vertices. Moreover, for a tree-colorable maximal planar graph of minimum degree 4 that contains exactly four odd-vertices, we show that the subgraph induced by its four odd-vertices is not a claw and contains no triangles.