Generalized Cages

Let 2 6 k1 < k2 < · · · < kt, 3 6 g1 < g2 < · · · < gs < N be integer parameters. A (k1, k2, . . . , kt; g1, g2, . . . , gs;N)-graph is a graph that contains vertices of degrees k1, k2, . . . , kt but no other degrees and cycles of lengths g1, g2, . . . , gs but no other cycles of length < N . For any given set of parameters satisfying the above conditions, we present an explicit construction of (k1, k2, . . . , kt; g1, g2, . . . , gs;N)-graphs and extend the concept of a cage (a smallest graph of given degree and girth) to that of a generalized cage – a smallest (k1, k2, . . . , kt; g1, g2, . . . , gs;N)-graph. We introduce several infinite families of generalized cages and study their basic properties in the context of connected, bipartite, and vertex-transitive graphs, as well as combinatorial configurations (in the context of multilaterals). ∗Dedicated to Branko Grünbaum whose work is a constant source of inspiration for us. †Supported in part by the project P1-0294. ‡Supported in part by the projects VEGA 1/0577/14, VEGA 1/0474/15, NSFC 11371307, and Project: Mobility ITMS code: 26110230082. §Supported in part by the projects P1-0294, J1-6720, N1-0011, and N1-0032. the electronic journal of combinatorics 22(1) (2015), #P1.77 1