The Vertex Arboricity of 1-Planar Graphs

The vertex arboricity a(G) of a graph G is the minimum number of colors required to color the vertices of G such that no cycle is monochromatic. A graph G is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we proved that every 1-planar graph without 5-cycles has minimum degree at most 5; Every 1-planar graph of girth at least 7 has minimum degree at most 3. The following conclusions can be obtained by combining the existing conclusions and our proofs: if G is a 1-planar graph without 5-cycles, then \(a(G)\le 3\) $a(G)\le 3$ ; if G is a 1-planar graph with \(g(G)\ge 7\) $g(G)\ge 7$ , then \(a(G)\le 2\) $a(G)\le 2$ .