Graphs without proper subgraphs of minimum degree 3 and short cycles

TL;DR

This paper investigates specific graphs with 2n-2 edges lacking certain subgraphs, disproves a conjecture about cycle lengths, and characterizes such graphs, advancing understanding of their structure and properties.

Contribution

It disproves a conjecture on cycle lengths in graphs with no proper subgraphs of minimum degree 3 and characterizes all such graphs with 2n-2 edges.

Findings

Disproved the conjecture that these graphs contain arbitrarily long cycles.

Characterized the structure of graphs with 2n-2 edges and no proper subgraph of minimum degree 3.

Resolved a related problem about leaf-to-leaf path lengths in trees.

Abstract

We study graphs on $n$ vertices which have $2n-2$ edges and no proper induced subgraphs of minimum degree $3$. Erd\H{o}s, Faudree, Gy\'arf\'as, and Schelp conjectured that such graphs always have cycles of lengths $3,4,5,\dots, C(n)$ for some function $C(n)$ tending to infinity. We disprove this conjecture, resolve a related problem about leaf-to-leaf path lengths in trees, and characterize graphs with $n$ vertices and $2n-2$ edges, containing no proper subgraph of minimum degree $3$.