Generalized Cages

Marko Boben, Robert Jajcay, Tomaz Pisanski


Let $2 \leq k_1 < k_2 < \ldots < k_t $, $3 \leq g_1 < g_2 < \ldots < g_s < N$ be integer parameters. A $(k_1,k_2,\ldots,k_t;g_1,g_2,\dots,g_s;N)$-graph is a graph that contains vertices of degrees $k_1,k_2,\ldots,k_t$ but no other degrees and cycles of lengths $g_1,g_2,\dots,g_s$ but no other cycles of length $< N$. For any given set of parameters satisfying the above conditions, we present an explicit construction of $(k_1,k_2,\ldots,k_t;g_1,g_2,\dots,g_s;N)$-graphs and extend the concept of a cage (a smallest graph of given degree and girth) to that of a generalized cage -- a smallest $(k_1,k_2,\ldots,k_t;g_1,g_2,\dots,g_s;N)$-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).


cages, vertex degrees, cycles

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