Covering the vertices of a graph by vertex-disjoint paths

Abstract

Define the path-covering number μ(G) of a finite graph G to be the minimum number of vertex-disjoint paths required to cover the vertices of G. Let g(n, k) be the minimum integer so that every graph, G, with n vertices and at least g(n, k) edges has μ(G) ≦ k. A relationship between μ(G) and the degree sequence for a graph G is found; this is used to show that (FORMULA PRESENTED) A further extremal problem is solved. © 1975 Pacific Journal of Mathematics.