Semicubic cages and small graphs of even girth from voltage graphs

A ({3, m}; g)-semicubic graph is a graph where the degree of each vertex is either 3 or m and the girth of the graph is g; if m = 3 we have a cubic graph. 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 of even girth g using voltage graphs over ℤ_(m). For girth g = 4t + 2, t ≥ 1, the constructed graphs have two vertices of degree m. For girth g = 4t, t ≥ 2, the construction produces graphs with exactly three vertices of degree m (of course, the remaining vertices are of degree 3 in both cases). In particular, we describe infinite families of ({3, m}; g)−semicubic graphs 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 of orders that match the best-known bounds for ({3, m}; g)−semicubic graphs until this moment.