On the order-diameter ratio of girth-diameter cages

TL;DR

The paper analyzes the order–diameter trade-off for girth-diameter cages $(k;g,d)$, introducing the asymptotic slope function $f(k,g)$ via $n(k;g,d) \le f(k,g)\,d + O_{k,g}(1)$. It proves sharp general bounds $\frac{M(k,g)}{g} \le f(k,g) \le \frac{n(k,g)}{g}$, while showing these bounds are not always tight, and shows that $f(k,g)$ is computable in $O_{k,g}(1)$ time using the notion of repeatable graphs. The authors determine $n(3;g,d)$ for $g \in \{4,5\}$ exactly and count corresponding cages, and they develop an exhaustive graph-generation algorithm that yields extensive data, including $n(k;g,d)$ values for hundreds of triples $(k,g,d)$ (with 107 new results). The largest cage resolved computationally is the $(3;7,35)$-cage of order $136$, and the work provides substantial new data and methods for the Cage Problem and the Degree Diameter Problem, including insights into bipartiteness for even girth cages and several open questions.

Abstract

For integers $k,g,d$, a $(k;g,d)$-cage (or simply girth-diameter cage) is a smallest $k$-regular graph of girth $g$ and diameter $d$ (if it exists). The order of a $(k;g,d)$-cage is denoted by $n(k;g,d)$. We determine asymptotic lower and upper bounds for the ratio between the order and the diameter of girth-diameter cages as the diameter goes to infinity. We also prove that this ratio can be computed in constant time for fixed $k$ and $g$. We theoretically determine the exact values $n(3;g,d)$, and count the number of corresponding girth-diameter cages, for $g \in \{4,5\}$. Moreover, we design and implement an exhaustive graph generation algorithm and use it to determine the exact order of several open cases and obtain -- often exhaustive -- sets of the corresponding girth-diameter cages. The largest case we generated and settled with our algorithm is a $(3;7,35)$-cage of order 136.

Figures (6)

• Figure 1: Illustration of the construction in \ref{['prop:upp_bnd']} for $(k,g)=(3,4)$ and $r=3$.
• Figure 2: The example showing that $f(4,7)<\frac{n(4,7)}{7}=\frac{67}{7}.$
• Figure 3: A graph that is repeatable with respect to $k=3$ and $g=4$ (where vertices in $N_0$ are shown larger).
• Figure 4: Removing a repeatable subgraph ($k=3$, $g=4$, $a=1$, $b=8$, $d=9$).
• Figure 5: Example of end block and gadget which are decoded by $133$ and $1221$.

Theorems & Definitions (27)

• theorem thmcountertheorem
• theorem thmcountertheorem
• proposition thmcounterproposition
• proof
• proposition thmcounterproposition
• proof
• corollary thmcountercorollary
• proposition thmcounterproposition
• proof
• proposition thmcounterproposition