Locally k-degenerate graphs: a definition and two conjectures.
A graph is called locally -regular if the neighbourhood of each vertex of induces a subgraph of which is regular of degree . We study graphs which are locally -regular and simultaneously regular of degree .
A subset of the vertex set of a graph is called locating-dominating, if for each there exists a vertex adjacent to and for any two distinct vertices , of the intersections of with the neighbourhoods of and are distinct. The maximum number of classes of a partition of whose classes are locating-dominating sets in is called the location-domatic number of Its basic properties are studied.
It is shown that every 3-connected planar graph with a large number of vertices has a long induced path.