Unions of 1-factors in $r$-graphs and overfull graphs

TL;DR

This paper establishes lower bounds on the edge coverage of $r$-graphs by $k$ 1-factors, introduces $k$-overfull-free $r$-graphs, and improves bounds for these classes, extending known results for cubic graphs.

Contribution

It provides new lower bounds for edge coverage in $r$-graphs and introduces the concept of $k$-overfull-free $r$-graphs with improved bounds.

Findings

Lower bounds for edge coverage in $r$-graphs by $k$ 1-factors.

Introduction of $k$-overfull-free $r$-graphs.

Enhanced bounds for $k$-overfull-free $r$-graphs.

Abstract

We prove lower bounds for the fraction of edges of an $r$-graph which can be covered by the union of $k$ 1-factors. The special case $r=3$ yields some known results for cubic graphs. Furthermore, we introduce the concept of $k$-overfull-free $r$-graphs and achieve better bounds for these graphs.