We give "if and only if" characterization of graphs with the following property: given n ≥ 3, edges of such graphs form matroids with circuits from the collection of all graphs with n fundamental cycles. In this way we refer to the notion of matroidal family defined by Simões-Pereira [2].
We consider circulant graphs having vertices, with prime. To any such graph we associate a certain number , that we call type of the graph. We prove that for the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.
The paper studies graphs in which each pair of vertices has exactly two common neighbours. It disproves a conjectury by P. Hliněný concerning these graphs.
We prove that for every number , the -iterated -path graph of is isomorphic to if and only if is a collection of cycles, each of length at least 4. Hence, is isomorphic to if and only if is a collection of cycles, each of length at least 4. Moreover, for we reduce the problem of characterizing graphs such that to graphs without cycles of length exceeding .
Two classes of graphs which are maximal with respect to the absence of Hamiltonian paths are presented. Block graphs with this property are characterized.
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.
For any and any , a graph is introduced. Vertices of are -tuples over and two -tuples are adjacent if they are in a certain relation. These graphs are graphs of a particular variant of the Tower of Hanoi problem. Namely, the graphs are isomorphic to the graphs of the Tower of Hanoi problem. It is proved that there are at most two shortest paths between any two vertices of . Together with a formula for the distance, this result is used to compute the distance between two vertices in...