Interval minors of complete bipartite graphs

TL;DR

This paper explores the maximum edges in bipartite graphs that avoid certain interval minors, providing exact results for small cases and bounds for larger ones, advancing understanding in graph minor theory.

Contribution

It determines exact maximum edges for $K_{2,s}$-interval minor free bipartite graphs and offers bounds and structural insights for $K_{3,s}$ cases.

Findings

Exact maximum edges for $K_{2,s}$-interval minor free graphs.

Bounds and structural properties for $K_{3,s}$-interval minor free graphs.

Abstract

Interval minors of bipartite graphs were recently introduced by Jacob Fox in the study of Stanley-Wilf limits. We investigate the maximum number of edges in $K_{r,s}$-interval minor free bipartite graphs. We determine exact values when $r=2$ and describe the extremal graphs. For $r=3$, lower and upper bounds are given and the structure of $K_{3,s}$-interval minor free graphs is studied.