On vertex Ramsey graphs with forbidden subgraphs

A graph is called H-free if it has no induced subgraph isomorphic to H. A graph is called $$N^i$$Ni-locally connected if $$G[\{ x\in V(G): 1\le d_G(w, x)\le i\}]$$G[{x?V(G):1≤dG(w,x)≤i}] is connected and $$N_2$$N2-locally connected if $$G[\{uv: \{uw, vw\...

Let P be a property of graphs. A graph G is vertex (P,k)-colourable if the vertex set V(G) of G can be partitioned into k sets V"1,V"2,...,V"k such that the subgraph G[V"i] of G belongs to P, i=1,2,...,k. If P is a hereditary property, then the set of ...

The vertex arboricity a(G) of a graph G is the minimum k such that V(G) can be partitioned into k sets where each set induces a forest. For a planar graph G, it is known that a(G)@?3. In two recent papers, it was proved that planar graphs without k-...