Given a family of graphs F , a graph G is F-saturated if no element of F is a subgraph of G, but for any edge e in G, some element of F is a subgraph of G + e. Let sat(n,F) denote the minimum numberâ€¦ (More)

Let D be a simple digraph without loops or digons. For any v âˆˆ V (D), the first out-neighborhood N+(v) is the set of all vertices with out-distance 1 from v and the second neighborhood N++(v) of v isâ€¦ (More)

We consider several constructions of edge critical 4-chromatic graphs which can be written as the union of a bipartite graph and a matching. In particular we construct such a graph G with each of theâ€¦ (More)

Let G be an infinite graph embedded in a closed 2-manifold, such that each open face of the embedding is homeomorphic to an open disk and is bounded by finite number of edges. For each vertex x of G,â€¦ (More)