Length bounds for cycle bases of graphs

The cycle space of a graph G is a vector space over GF(2) which is formed by all Eulerian subgraphs of G with vector addition X⊕Y:=(X∪Y)∖(X∩Y) and scalar multiplication 1⋅X=X, 0⋅X=∅. A base of this vector space is called a cycle base. A cycle base is used to examine the cyclic structure of a graph. The length of a cycle base is the number of its edges in the base. A minimum cycle base is that having the least number of edges. In this paper, we study the length bounds for cycle bases and the minimum cycle bases. A complete characterization is given for a 2-connected graph G with a cycle base of length 2|E(G)|−|V(G)| (it is a lower bound obtained by Leydold and Stadler (1998) [11]). In addition, we derive a sharp lower length bound for minimum cycle bases. As for upper bounds, Horton (1987) showed that the length of a minimum cycle base of a graph with n vertices is at most 3(n−1)(n−2)/2. We improve the bound substantially for graphs on the projective plane where it is at most ⌊13n/2⌋−9.