Semicubic cages and small graphs of even girth from voltage graphs

TL;DR

This paper introduces new methods to construct semicubic graphs with even girth and small order, using generalized cage gluing and voltage graph techniques, producing infinite families with optimal properties for certain girths.

Contribution

It presents novel construction techniques for semicubic graphs of even girth, including voltage graph methods over cyclic groups, expanding known families and bounds.

Findings

Constructed infinite families of semicubic graphs for girths 6, 8, 10, 12.

Produced graphs with optimal known bounds for girths 10 and 12.

Included unique cages for girths 6 and 8 when m=3.

Abstract

An \emph{$(3,m;g)$ semicubic graph} is a graph in which all vertices have degrees either $3$ or $m$ and fixed girth $g$. In this paper, we construct families of semicubic graphs of even girth and small order using two different techniques. The first technique generalizes a previous construction which glues cubic cages of girth $g$ together at remote vertices (vertices at distance at least $g/2$). The second technique, the main content of this paper, produces bipartite semicubic $(3,m; g)$-graphs with fixed even girth $g = 4t$ or $4t+2$ using voltage graphs over $\mathbb{Z}_{m}$. When $g = 4t+2$, the graphs have two vertices of degree $m$, while when $g = 4t$ they have exactly three vertices of degree $m$ (the remaining vertices are of degree $3$ in both cases). Specifically, we describe infinite families of semicubic graphs $(3,m; g)$ for $g = \{6, 8, 10, 12\}$ for infinitely many values of $m$. The cases $g = \{6,8\}$ include the unique $6$-cage and the unique $8$-cage when $m = 3$. The families obtained in this paper for girth $g=\{10,12\}$ include examples with the best known bounds for semicubic graphs $(3,m; g)$