Graphs and Betweeness
Graphs in which Every Finite Path is Contained in a Circuit.
Graphs in which every path is contained in a Hamilton path.
Graphs maximal with respect to hom-properties
For a simple graph H, →H denotes the class of all graphs that admit homomorphisms to H (such classes of graphs are called hom-properties). We investigate hom-properties from the point of view of the lattice of hereditary properties. In particular, we are interested in characterization of maximal graphs belonging to →H. We also provide a description of graphs maximal with respect to reducible hom-properties and determine the maximum number of edges of graphs belonging to →H.
Graphs minimal with respect to contractions in some subfamilies of maximal planar graphs
Graphs related to diameter and center.
Graphs with extremal Randić index when the minimum degree of vertices is two
Graphs with greatest number of matchings.
Graphs with many copies of a given subgraph.
Graphs with maximum and minimum independence numbers.
Graphs with prescribed maximal subgraphs and critical chromatic graphs
Graphs with prescribed size and vertex parities
Graphs with rainbow connection number two
An edge-coloured graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colours. The rainbow connection number of a connected graph G, denoted rc(G), is the smallest number of colours that are needed in order to make G rainbow connected. In this paper we prove that rc(G) = 2 for every connected graph G of order n and size m, where . We also characterize graphs with rainbow connection number two and large clique number.
Graphs with smallest sum of squares of vertex degrees
-free graphs of large minimum degree.
Hadwiger numbers of finite graphs
Hamiltonian circuits and path coverings of vertices in graphs
Hamiltonian circuits in cubic graphs
Hamiltonian Lines in Infinite Graphs with Few Verticles of Small Valency
Hamiltonian shortage, path partitions of vertices, and matchings in a graph