On Maximal Circuits in Finite Graphs

Publisher Summary If the valency d(v) of each vertex v of a graph G is at least 1/2n(G), where n(G) is the number of vertices of G and n(G) 2 , then G allows a hamiltonian circuit—that is, a circuit, which contains every vertex of G. This chapter focuses on the following refinement: If d(v) + d(w) ≥ n(G) > 2 for any two different, nonadjacent vertices v, w of G, then G contains a hamiltonian circuit. Circuits and paths in a graph G are considered as subgraphs of G. A circuit C in G is called maximal if there exists no circuit C' such that V(C') ⊃ V(C). A maximal circuit C with a direction of transversing and a component H of G – (C) are fixed.