Minimum $k$-critical-bipartite graphs: the irregular Case

TL;DR

This paper investigates the structure of minimum $k$-critical-bipartite graphs, extending previous biregular results to irregular cases, and establishes tight bounds on their connectivity.

Contribution

It generalizes the characterization of minimum $k$-critical-bipartite graphs to irregular bipartite graphs and provides tight lower bounds on their connectivity.

Findings

Extended results to irregular bipartite graphs.

Proved tight lower bounds on connectivity.

Characterized the structure of minimum $k$-critical-bipartite graphs.

Abstract

We study the problem of finding a minimum $k$-critical-bipartite graph of order $(n,m)$: a bipartite graph $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and $n>m>1$, which is $k$-critical-bipartite, and the tuple $(|E|, \Delta_U, \Delta_V)$, where $\Delta_U$ and $\Delta_V$ denote the maximum degree in $U$ and $V$, respectively, is lexicographically minimum over all such graphs. $G$ is $k$-critical-bipartite if deleting at most $k=n-m$ vertices from $U$ yields $G'$ that has a complete matching, i.e., a matching of size $m$. Cichacz and Suchan solved the problem for biregular bipartite graphs. Here, we extend their results to bipartite graphs that are not biregular. We also prove tight lower bounds on the connectivity of $k$-critical-bipartite graphs.