Extremal $K_4$-minor-free graphs without short cycles

TL;DR

This paper characterizes the maximum edges in $K_4$-minor-free graphs with specified girth, revealing multiple extremal configurations depending on parity of girth and number of vertices.

Contribution

It provides exact extremal edge counts for $K_4$-minor-free graphs with girth 5 or even, and discusses diversity of extremal graphs for odd girth and even vertices.

Findings

Maximum edges for girth 5 and even girth cases determined.

Multiple extremal graphs exist for odd girth and even vertices.

Structural properties of extremal graphs analyzed.

Abstract

We determine the maximum number of edges in a $K_4$-minor-free $n$-vertex graph of girth $g$, when $g = 5$ or $g$ is even. We argue that there are many different $n$-vertex extremal graphs, if $n$ is even and $g$ is odd.