On the maximum number of independent edges in cubic graphs

Abstract

Let the reals be extended to include oo with o~ > r for every real nuraber r. Given an extended real number r, a property P(r) of graphs is super-hereditary if, whenever graph G has property P(r) and H is a subgraph of G, then H has property P(s) with s I> r. Notice that girth is a super-hereditary property if a forest has girth oo. In this paper, we prove that, given a super-hereditary proper~y P of graphs, if g is the smallest order of a graph with property P and having one vertex of degree 2 and all others of degree 3, and if G is a cubic graph with n vertices and property P, then G contains a matching with at least 1⁄2n(3g1)/(3g-~1) edges. In the case that P is the property of having girth r we describe all graphs having exactly the maximum matching size named. Our result has the corollary that every cubic graph with n vertices has a matching containing at least ~ n edges. Let L be a graph with super-hereditary property P in which one vertex has degree two and all others have degree 3 and such that the the order g of L is as s~nall as possible. For example, if P is 'having girth 3', L is /(4 with one edge subdivided and g = 5. If P is 'having girth 4', then L i,; K3.3 with one edge subdivided, and g = 7. The existence of L in the case that P is 'having girth r' ~.s shown by observing that an r-cage with one edge subdivided will serve (see [3, Theorem 8.82], for a proof that r-cages exist for all r>~3). Bollob~is and Eldridge [2], by the same method used in this paper, obtained the greatest lower bound for the size of a maximum matching in a k-connected (or Aedge-connected) graph with a given number of vertices anzl given minimal and maximal degrees. In this paper we find the greatest lower bound for the size of a maximum matching in a cubic graph with n vertices and with sup,~r-hereditary property P(r). In particular: