On $(k,g)$-Graphs without $(g+1)$-Cycles

TL;DR

This paper investigates the smallest $k$-regular graphs with girth $g$ that lack cycles of length $g+1$, establishing bounds, properties, and algorithms to identify such graphs, contributing to the Girth Pair and Cage problems.

Contribution

It introduces new bounds, proves monotonicity of the order function, and develops algorithms for constructing minimal $(k,g, ext{g+1})$-graphs.

Findings

Established monotonicity of $n(k,g, ext{g+1})$ with respect to $g$

Derived universal lower bounds for $n(k,g, ext{g+1})$

Developed algorithms to generate and identify minimal such graphs

Abstract

A $(k,g,\underline{g+1})$-graph is a $k$-regular graph of girth $g$ which does not contain cycles of length $g+1$. Such graphs are known to exist for all parameter pairs $k \geq 3, g \geq 3 $, and we focus on determining the orders $n(k,g,\underline{g+1})$ of the smallest $(k,g,\underline{g+1})$-graphs. This problem can be viewed as a special case of the previously studied Girth Pair Problem, the problem of finding the order of a smallest $k$-regular graph in which the length of a smallest even length cycle and the length of a smallest odd length cycle are prescribed. When considering the case of an odd girth $g$, this problem also yields results towards the Cage Problem, the problem of finding the order of a smallest $k$-regular graph of girth $g$. We establish the monotonicity of the function $n(k,g,\underline{g+1})$ with respect to increasing $g$, and present universal lower bounds for the values $n(k,g,\underline{g+1})$. We propose an algorithm for generating all $(k,g,\underline{g+1})$-graphs on $n$ vertices, use this algorithm to determine several of the smaller values $n(k,g,\underline{g+1})$, and discuss various approaches to finding smallest $(k,g,\underline{g+1})$-graphs within several classes of highly symmetrical graphs.