An Approach to the Girth Problem in Cubic Graphs

We offer a new, gradual approach to the largest girth problem for cubic graphs. It is easily observed that the largest possible girth of all (Formula presented.) -vertex cubic graphs is attained by a 2-connected graph (Formula presented.). By Petersen's graph theorem, (Formula presented.) is the disjoint union of a 2-factor and a perfect matching (Formula presented.). We refer to the edges of (Formula presented.) as ribs and classify the cycles in (Formula presented.) by their number of ribs. We define (Formula presented.) to be the smallest integer (Formula presented.) such that every cubic (Formula presented.) -vertex graph with a given perfect matching (Formula presented.) has a cycle of length at most (Formula presented.) with at most (Formula presented.) ribs. Here, we determine this function up to small additive constant for (Formula presented.) and up to a small multiplicative constant for larger (Formula presented.).