Every planar graph with cycles of length neither 4 nor 5 is \((1,1,0)\)-colorable
Let \(d_1, d_2,\dots ,d_k\) be \(k\) non-negative integers. A graph \(G\) is \((d_1,d_2,\ldots ,d_k)\)-colorable, if the vertex set of \(G\) can be partitioned into subsets \(V_1,V_2,\ldots ,V_k\) such that the subgraph \(G[V_i]\) induced by \(V_i\) has maximum degree at most \(d_i\) for \(i=1,2,\ldots ,k.\) Let \(\digamma \) be the family of planar graphs with cycles of length neither 4 nor 5. Steinberg conjectured that every graph of \(\digamma \) is \((0,0,0)\)-colorable. In this paper, we prove that every graph of \(\digamma \) is \((1,1,0)\)-colorable.
planar graph steinbergs conjecture improper coloring