ABSTRACT

All graphs considered will be finite and may contain multiple edges but no loops. The main purpose is to obtain bounds on the length of a longest cycle in a 3-connected graph of maximum degree. In this chapter, the authors provide the introduction to the longest cycles in 3-connected graphs of bounded maximum degree. They also provide definitions and preliminary lemmas of the 3-connected graphs of bounded maximum degree. The authors adopt the notation and terminology of Tutte with the exception that they will refer to 'polygons', simple paths', and 'valency' as cycles, paths and degree respectively. Cleavage units are the minimal cleavage graphs obtained by recursively constructing cleavage graphs from cleavage graphs; they form a tree called the cleavage unit tree of graph in a manner analagous to the block/cut-vertex tree of a graph.