On Independent Circuits Contained in a Graph

A family of circuits of a graph G is said to be independent if no two of the circuits have a common vertex ; it is called edge-independent if no two of them have an edge in common . A set of vertices will be called a representing set for the circuits (for the sake of brevity we shall call it a representing set), if every circuit of G passes through at least one vertex of the representing set . Denote by l(G) = k the maximum number of circuits in an independent family and by R(G) the minimum number of vertices of a representing set . Dirac and Gallai asked whether there is any relation between I (G) and R (G) (trivially R(G) > I(G)) . B . Bollobás (unpublished) proved that if l(G) = 1, then R(G) < 3 and the coniplete graph of five vertices shows that R(G) < 3 is best possible . Consider now all graphs with I(G) = k . Denote by r(k) the maximum value of R(G) for all graphs with I(G) = k . It is not immediately obvious that r(k) is finite and the theorem of Bollobás states that r(1) = 3 . The value of r(2) does not seem to be known . We are going to prove the following